{"id":41131,"date":"2022-04-05T15:02:28","date_gmt":"2022-04-05T09:32:28","guid":{"rendered":"https:\/\/ccssmathanswers.com\/?p=41131"},"modified":"2022-04-07T11:11:10","modified_gmt":"2022-04-07T05:41:10","slug":"big-ideas-math-geometry-answers-chapter-6","status":"publish","type":"post","link":"https:\/\/bigideasmathanswers.com\/big-ideas-math-geometry-answers-chapter-6\/","title":{"rendered":"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles"},"content":{"rendered":"<p>Give your kid the right assistance he\/she needs regarding the Big Ideas Math Geometry Chapter 6 Relationships within Triangles concepts. Practicing from the <a href=\"https:\/\/bigideasmathanswers.com\/big-ideas-math-geometry-answers\/\">Big Ideas Math Geometry Answers<\/a> Ch 6 Relationships within Triangles on a frequent basis help you to attempt the exam with confidence. You can avail these handy resources for BIM Book Geometry Chapter 6 Relationships within Triangles Solution Key and clear the exams with high scores. In fact, you can download the BIM Book Geometry Ch 6 Relationships within Triangles Answers for free of cost and prepare them offline too.<\/p>\n<h2>Big Ideas Math Book Geometry Answer Key Chapter 6 Relationships Within Triangles<\/h2>\n<p>Have a deeper understanding of concepts from the surface level by making the most out of these BIM Geometry Solutions for Chapter 6. Solve the Questions within the BigIdeas Math Geometry Textbook Answer Key PDF on a frequent basis and answer any kind of question in the exam easily. Just click on the Topicwise Big Ideas Math Geometry Ch 6 Relationships within Triangles Answers and kickstart your preparation.<\/p>\n<ul>\n<li><a href=\"#Relationships_Within_Triangles_Maintaining_Mathematical_Proficiency\">Relationships Within Triangles Maintaining Mathematical Proficiency &#8211; Page 299<\/a><\/li>\n<li><a href=\"#Relationships_Within_Triangles_Mathematical_Practices\">Relationships Within Triangles Mathematical Practices &#8211; Page 300<\/a><\/li>\n<li><a href=\"#6.1_Perpendicular_And_Angle_Bisectors\">6.1 Perpendicular And Angle Bisectors &#8211; Page 301<\/a><\/li>\n<li><a href=\"#Lesson_6.1_Perpendicular_And_Angle_Bisectors\">Lesson 6.1 Perpendicular And Angle Bisectors &#8211; Page(302-308)<\/a><\/li>\n<li><a href=\"#Exercise_6.1_Perpendicular_And_Angle_Bisectors\">Exercise 6.1 Perpendicular And Angle Bisectors &#8211; Page(306-308)<\/a><\/li>\n<li><a href=\"#6.2_Bisectors_of_Triangles\">6.2 Bisectors of Triangles &#8211; Page 309<\/a><\/li>\n<li><a href=\"#Lesson_6.2_Bisectors_of_Triangles\">Lesson 6.2 Bisectors of Triangles &#8211; Page(310-318)<\/a><\/li>\n<li><a href=\"#Exercise_6.2_Bisectors_of_Triangles\">Exercise 6.2 Bisectors of Triangles &#8211; Page(315-318)<\/a><\/li>\n<li><a href=\"#6.3_Medians_and_Altitudes_of_Triangles\">6.3 Medians and Altitudes of Triangles &#8211; Page 319<\/a><\/li>\n<li><a href=\"#Lesson_6.3_Medians_and_Altitudes_of_Triangles\">Lesson 6.3 Medians and Altitudes of Triangles &#8211; Page(320-326)<\/a><\/li>\n<li><a href=\"#Exercise_6.3_Medians_and_Altitudes_of_Triangles\">Exercise 6.3 Medians and Altitudes of Triangles &#8211; Page(324-326)<\/a><\/li>\n<li><a href=\"#6.1_and_6.3_Quiz\">6.1 and 6.3 Quiz &#8211; Page 328<\/a><\/li>\n<li><a href=\"#6.4_The_Triangle_Midsegment_Theorem\">6.4 The Triangle Midsegment Theorem &#8211; Page 329<\/a><\/li>\n<li><a href=\"#Lesson_6.4_The_Triangle_Midsegment_Theorem\">Lesson 6.4 The Triangle Midsegment Theorem &#8211; Page (330-334)<\/a><\/li>\n<li><a href=\"#Exercise_6.4_The_Triangle_Midsegment_Theorem\">Exercise 6.4 The Triangle Midsegment Theorem &#8211; Page(333-334)<\/a><\/li>\n<li><a href=\"#6.5_Indirect_Proof_and_Inequalities_in_One_Triangle\">6.5 Indirect Proof and Inequalities in One Triangle &#8211; Page 335<\/a><\/li>\n<li><a href=\"#Lesson_6.5_Indirect_Proof_and_Inequalities_in_One_Triangle\">Lesson 6.5 Indirect Proof and Inequalities in One Triangle &#8211; Page(336-342)<\/a><\/li>\n<li><a href=\"#Exercise_6.5_Indirect_Proof_and_Inequalities_in_One_Triangle\">Exercise 6.5 Indirect Proof and Inequalities in One Triangle &#8211; Page(340-342)<\/a><\/li>\n<li><a href=\"#6.6_Inequalities_in_Two_Triangles\">6.6 Inequalities in Two Triangles &#8211; Page 343<\/a><\/li>\n<li><a href=\"#Lesson_6.6_Inequalities_in_Two_Triangles\">Lesson 6.6 Inequalities in Two Triangles &#8211; Page(344-348)<\/a><\/li>\n<li><a href=\"#Exercise_6.6_Inequalities_in_Two_Triangles\">Exercise 6.6 Inequalities in Two Triangles &#8211; Page(347-348)<\/a><\/li>\n<li><a href=\"#Relationships_Within_Triangles_Chapter_Review\">Relationships Within Triangles Chapter Review &#8211; Page(350-352)<\/a><\/li>\n<li><a href=\"#Relationships_Within_Triangles_Chapter_Test\">Relationships Within Triangles Chapter Test &#8211; Page 353<\/a><\/li>\n<li><a href=\"#Relationships_Within_Triangles_Cumulative_Assessment\">Relationships Within Triangles Cumulative Assessment &#8211; Page(354-355)<\/a><\/li>\n<\/ul>\n<h3><a id=\"Relationships_Within_Triangles_Maintaining_Mathematical_Proficiency\"><\/a>Relationships Within Triangles Maintaining Mathematical Proficiency<\/h3>\n<p>Write an equation of the line passing through point P that is perpendicular to the given line.<\/p>\n<p>Question 1.<br \/>\nP(3, 1), y = \\(\\frac{1}{3}\\)x &#8211; 5<\/p>\n<p>Answer:<br \/>\ny = -3x + 10<\/p>\n<p>Explanation:<br \/>\nThe slope of the given line is \\(\\frac { 1 }{ 3 } \\). Since the slope of the perpendicular line must be the negative reciprocal of the slope of the given line.<br \/>\nThe slope of the perpendicular line = -3<br \/>\nSubstitute the values in y = mx + c<br \/>\n1 = -3(3) + c<br \/>\n1 = -9 + c<br \/>\nc = 1 + 9<br \/>\nc = 10<br \/>\nuse the slope intercept form of a linear equation again substitute m, c<br \/>\ny = -3x + 10<\/p>\n<p>Question 2.<br \/>\nP(4, &#8211; 3), y = &#8211; x &#8211; 5<\/p>\n<p>Answer:<br \/>\ny = \\(\\frac { 1 }{ 3 } \\)x &#8211; \\(\\frac { 5 }{ 3 } \\)<\/p>\n<p>Explanation:<br \/>\nThe slope of the given line is -3. Since the slope of the perpendicular line must be the negative reciprocal of the slope of the given line.<br \/>\nThe slope of the perpendicular line = \\(\\frac { 1 }{ 3 } \\)<br \/>\nSubstitute the values in y = mx + c<br \/>\n-3 = &#8211;\\(\\frac { 1 }{ 3 } \\)(4) + c<br \/>\nc = -3 + \\(\\frac { 4 }{ 3 } \\) = \\(\\frac { -9 + 4 }{ 3 } \\) = \\(\\frac { -5 }{ 3 } \\)<br \/>\nuse the slope intercept form of a linear equation again substitute m, c<br \/>\ny = \\(\\frac { 1 }{ 3 } \\)x + \\(\\frac { -5 }{ 3 } \\)<br \/>\ny = \\(\\frac { 1 }{ 3 } \\)x &#8211; \\(\\frac { 5 }{ 3 } \\)<\/p>\n<p>Question 3.<br \/>\nP(- 1, &#8211; 2), y = &#8211; 4x + 13<\/p>\n<p>Answer:<br \/>\ny = \\(\\frac { 1 }{ 4 } \\)x &#8211; \\(\\frac { 7 }{ 4 } \\)<\/p>\n<p>Explanation:<br \/>\nThe slope of the given line is -4. Since the slope of the perpendicular line must be the negative reciprocal of the slope of the given line.<br \/>\nThe slope of the perpendicular line = \\(\\frac { 1 }{ 4 } \\)<br \/>\nSubstitute the values in y = mx + c<br \/>\n-2 = \\(\\frac { 1 }{ 4 } \\)(-1) + c<br \/>\nc = -2 + \\(\\frac { 1 }{ 4 } \\) = \\(\\frac { -8 + 1 }{ 4 } \\) = \\(\\frac { -7 }{ 4 } \\)<br \/>\nuse the slope intercept form of a linear equation again substitute m, c<br \/>\ny = \\(\\frac { 1 }{ 4 } \\)x + \\(\\frac { -7 }{ 4 } \\)<br \/>\ny = \\(\\frac { 1 }{ 4 } \\)x &#8211; \\(\\frac { 7 }{ 4 } \\)<\/p>\n<p>Write the sentence as an inequality.<\/p>\n<p>Question 4.<br \/>\nA number w is at least &#8211; 3 and no more than 8.<\/p>\n<p>Answer:<br \/>\n-3 \u2264 w &lt; 8<\/p>\n<p>Explanation:<br \/>\nAt least means \u2265 and no more than means &lt;<br \/>\nw \u2265 -3 and w &lt; 8<br \/>\n-3 \u2264 w &lt; 8<\/p>\n<p>Question 5.<br \/>\nA number m is more than 0 and less than 11.<\/p>\n<p>Answer:<br \/>\n0 &lt; m &lt; 11<\/p>\n<p>Explanation:<br \/>\nmore than means &gt; and less than means &lt;<br \/>\nm &gt; 0 and m &lt; 11<br \/>\n0 &lt; m &lt; 11<\/p>\n<p>Question 6.<br \/>\nA number s is less than or equal to 5 or greater than 2.<\/p>\n<p>Answer:<br \/>\n2 &lt; s \u2264 5<\/p>\n<p>Explanation:<br \/>\nless than or equal to means \u2264 and greater than means &gt;<br \/>\ns \u2264 5 or s &gt; 2<br \/>\n2 &lt; s \u2264 5<\/p>\n<p>Question 7.<br \/>\nA number d is fewer than 12 or no less than &#8211; 7.<\/p>\n<p>Answer:<br \/>\n-7 &lt; d &lt; 12<\/p>\n<p>Explanation:<br \/>\nfewer than means &lt; no less than means &gt;<br \/>\nd &lt; 12 or d &gt; -7<br \/>\n-7 &lt; d &lt; 12<\/p>\n<p>Question 8.<br \/>\n<strong>ABSTRACT REASONING<\/strong><br \/>\nIs it possible for the solution of a compound inequality to be all real numbers? Explain your reasoning.<\/p>\n<p>Answer:<br \/>\nYes<\/p>\n<h3><a id=\"Relationships_Within_Triangles_Mathematical_Practices\"><\/a>Relationships Within Triangles Mathematical Practices<\/h3>\n<p><strong>Monitoring Progress<\/strong><\/p>\n<p>Refer to the figures at the top of the page to describe each type of line, ray, or segment in a triangle.<\/p>\n<p>Question 1.<br \/>\nperpendicular bisector<\/p>\n<p>Answer:<br \/>\nThe perpendicular bisector of a side of a triangle is a line perpendicular to the side and passes through its midpoint. Three perpendicular bisectors of sides meet at a point in a triangle.<\/p>\n<p>Question 2.<br \/>\nangle bisector<\/p>\n<p>Answer:<br \/>\nThe angle bisector of an angle in a triangle is a straight line that divides the angle into two congruent angles.<\/p>\n<p>Question 3.<br \/>\nmedian<\/p>\n<p>Answer:<br \/>\nIt is a line segment from a vertex to the midpoint of the opposite side.<\/p>\n<p>Question 4.<br \/>\naltitude<\/p>\n<p>Answer:<br \/>\nIt is a segment from a vertex of the triangle, perpendicular to the side opposite that vertex of the triangle.<\/p>\n<p>Question 5.<br \/>\nmidsegment<\/p>\n<p>Answer:<br \/>\nIt is a segment that connects the mid points of two sides in a triangle.<\/p>\n<h3><a id=\"6.1_Perpendicular_And_Angle_Bisectors\"><\/a>6.1 Perpendicular and Angle Bisectors<\/h3>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-41305\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-1-1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 1\" width=\"416\" height=\"238\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-1-1.png 416w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-1-1-300x172.png 300w\" sizes=\"(max-width: 416px) 100vw, 416px\" \/><\/p>\n<p><strong>Exploration 1<\/strong><\/p>\n<p>Points on a Perpendicular Bisector<\/p>\n<p>Work with a partner. Use dynamic geometry software.<\/p>\n<p>a. Draw any segment and label it \\(\\overline{A B}\\). ConStruct the perpendicular bisector of \\(\\overline{A B}\\).<br \/>\nAnswer:<\/p>\n<p>b. Label a point C that is on the perpendicular bisector of \\(\\overline{A B}\\) but is not on \\(\\overline{A B}\\).<br \/>\nAnswer:<\/p>\n<p>c. Draw \\(\\overline{C A}\\) and \\(\\overline{C B}\\) and find their lengths. Then move point C to other locations on the perpendicular bisector and note the lengths of \\(\\overline{C A}\\) and \\(\\overline{C B}\\).<br \/>\nAnswer:<\/p>\n<p>d. Repeat parts (a) &#8211; (c) with other segments. Describe any relationships(s) you notice.<br \/>\nAnswer:<\/p>\n<p><strong>Exploration 2<\/strong><\/p>\n<p>Points on an Angle Bisector<\/p>\n<p>Work with a partner. Use dynamic geometry software.<\/p>\n<p>a. Draw two rays \\(\\vec{A}\\)B and \\(\\vec{A}\\)C to form \u2220BAC. Construct the bisector of \u2220BAC.<br \/>\nAnswer:<\/p>\n<p>b. Label a point D on the bisector of \u2220BAC.<br \/>\nAnswer:<\/p>\n<p>c. Construct and find the lengths of the perpendicular segments from D to the sides of \u2220BAC. Move point D along the angle bisector and note how the lengths change.<br \/>\nAnswer:<\/p>\n<p>d. Repeat parts (a)-(c) with other angles. Describe an relationship(s) you notice.<br \/>\n<strong>USING TOOLS STRATEGICALLY<\/strong><br \/>\nTo be proficient in math, you need to visualize the results of varying assumptions, explore consequences, and compare predictions with data.<br \/>\nAnswer:<\/p>\n<p>Communicate Your Answer<\/p>\n<p>Question 3.<br \/>\nWhat conjectures can you make about a point on the perpendicular bisector of a segment and a point on the bisector of an angle?<br \/>\nAnswer:<\/p>\n<p>Question 4.<br \/>\nIn Exploration 2. what is the distance from point D to \\(\\vec{A}\\)B when the distance from D to \\(\\vec{A}\\)C is 5 units? Justify your answer.<br \/>\nAnswer:<\/p>\n<h3><a id=\"Lesson_6.1_Perpendicular_And_Angle_Bisectors\"><\/a>Lesson 6.1 Perpendicular and Angle Bisectors<\/h3>\n<p>Use the diagram and tile given information to find the indicated measure.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-41304\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-2.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 2\" width=\"129\" height=\"168\" \/><\/p>\n<p>Question 1.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41303\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-3.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 3\" width=\"32\" height=\"23\" \/> is the perpendicular bisector of \\(\\overline{W Y}\\), and \\(\\overline{y Z}\\) = 13.75. Find WZ.<\/p>\n<p>Answer:<br \/>\nWZ = 13.75<\/p>\n<p>Explanation:<br \/>\nUsing the perpendicular bisector theorem YZ = WZ<br \/>\n13.75 = WZ<\/p>\n<p>Question 2.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41303\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-3.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 3\" width=\"32\" height=\"23\" \/> is the perpendicular bisector of \\(\\overline{W Y}\\), WZ = 4n &#8211; 13, and YZ = n + 17. Find YZ.<\/p>\n<p>Answer:<br \/>\nYZ = 27<\/p>\n<p>Explanation:<br \/>\nUsing the perpendicular bisector theorem YZ = WZ<br \/>\nn + 17 = 4n &#8211; 13<br \/>\n4n &#8211; n = 17 + 13<br \/>\n3n = 30<br \/>\nn = 10<br \/>\nYZ = 10 + 17 = 27<\/p>\n<p>Question 3.<br \/>\nFind WX when WZ = 20.5. WY = 14.8. and YZ = 20.5.<\/p>\n<p>Answer:<br \/>\nWX = 7.4<\/p>\n<p>Explanation:<br \/>\nBecause WZ = ZY and ZX is the perpendicular bisector of WY by the converse of perpendicular bisector theorem. By the definition of segment sector, WY = 2WX<br \/>\n14.8 = 2WX<br \/>\nWX = 7.4<\/p>\n<p>Use the diagram and the given information to find the indicated measure.<\/p>\n<p>Question 4.<br \/>\n\\(\\vec{B}\\)D bisects \u2220ABC, and DC = 6.9, Find DA.<\/p>\n<p>Answer:<br \/>\nDA = 6.9<\/p>\n<p>Explanation:<br \/>\nBy using the angle bisector theorem<br \/>\nDC = DA<br \/>\nDA = 6.9<\/p>\n<p>Question 5.<br \/>\n\\(\\vec{B}\\)D bisects \u2220ABC, AD = 3z + 7, and CD = 2z + 11. Find CD.<\/p>\n<p>Answer:<br \/>\nz = 4<\/p>\n<p>Explanation:<br \/>\nBy using the angle bisector theorem<br \/>\nDC = DA<br \/>\n2z + 11 = 3z + 7<br \/>\n3z &#8211; 2z = 11 &#8211; 7<br \/>\nz = 4<\/p>\n<p>Question 6.<br \/>\nFind m\u2220ABC when AD = 3.2, CD = 3.2, and m\u2220DBC = 39\u00b0.<\/p>\n<p>Answer:<br \/>\nm\u2220DBC = 78\u00b0<\/p>\n<p>Explanation:<br \/>\nm\u2220DBC = 39 = m\u2220DBA<br \/>\nm\u2220ABC = m\u2220DBC + m\u2220DBA<br \/>\n= 39 + 39 = 78\u00b0<\/p>\n<p>Question 7.<br \/>\nDo you have enough information to conclude that \\(\\vec{Q}\\)S bisects \u2220PQR? Explain.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41302\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-4.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 4\" width=\"205\" height=\"94\" \/><br \/>\nAnswer:<br \/>\nYes,<br \/>\nBecause PS = SR.<\/p>\n<p>Question 8.<br \/>\nWrite an equation of the perpendicular bisector of the segment with endpoints (- 1, &#8211; 5) and (3, &#8211; 1).<\/p>\n<p>Answer:<br \/>\ny + 5 = 0<\/p>\n<p>Explanation:<br \/>\nMidpoint = (\\(\\frac { -1 + 3 }{ 2 } \\), \\(\\frac { -5 &#8211; 1 }{ 2 } \\))<br \/>\n= (\\(\\frac { 2 }{ 2 } \\), \\(\\frac { -6 }{ 2 } \\)) = (1, -3)<br \/>\nSlope of the line = \\(\\frac { -1 + 1 }{ 3 + 5 } \\) = 0<br \/>\nSlope of the perpendicular line = 0<br \/>\nEquation of the perpendicular bisector is (y + 5) = 0(x + 1)<br \/>\ny + 5 = 0<\/p>\n<h3><a id=\"Exercise_6.1_Perpendicular_And_Angle_Bisectors\"><\/a>Exercise 6.1 Perpendicular and Angle Bisectors<\/h3>\n<p>Vocabulary and Core Concept Check<\/p>\n<p>Question 1.<br \/>\n<strong>COMPLETE THE SENTENCE<\/strong><br \/>\nPoint C is in the interior of \u2220DEF. If \u2220DEC and \u2220CEF are congruent, then \\(\\vec{E}\\)C is the ________ of \u2220DEF.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48930\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 1\" width=\"558\" height=\"60\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-1.png 558w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-1-300x32.png 300w\" sizes=\"(max-width: 558px) 100vw, 558px\" \/><\/p>\n<p>Question 2.<br \/>\n<strong>DIFFERENT WORDS, SAME QUESTION<\/strong><br \/>\nWhich is different? Find \u201cboth\u201d answers.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-41301\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-5.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 5\" width=\"216\" height=\"156\" \/><\/p>\n<p>Is point B the same distance from both X and Z?<\/p>\n<p>Answer:<br \/>\nYes.<\/p>\n<p>Is point B equidistant from X and Z?<\/p>\n<p>Answer:<br \/>\nYes<\/p>\n<p>Is point B collinear with X and Z?<\/p>\n<p>Answer:<br \/>\nNo.<\/p>\n<p>Is point B on the perpendicular bisector of \\(\\overline{X Z}\\)?<\/p>\n<p>Answer:<br \/>\nYes<\/p>\n<p>Monitoring Progress and Modeling with Mathematics<\/p>\n<p>In Exercises 3-6, find the indicated measure. Explain your reasoning.<\/p>\n<p>Question 3.<br \/>\nGH<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41300\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6\" width=\"137\" height=\"134\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48931\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-3.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 3\" width=\"547\" height=\"97\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-3.png 547w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-3-300x53.png 300w\" sizes=\"(max-width: 547px) 100vw, 547px\" \/><\/p>\n<p>Question 4.<br \/>\nQR<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41299\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-7.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 7\" width=\"130\" height=\"120\" \/><\/p>\n<p>Answer:<br \/>\nQR = RS<br \/>\nSo, QR = 1.3<\/p>\n<p>Question 5.<br \/>\nAB<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41298\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-8.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 8\" width=\"153\" height=\"104\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48932\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-5.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 5\" width=\"596\" height=\"246\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-5.png 596w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-5-300x124.png 300w\" sizes=\"(max-width: 596px) 100vw, 596px\" \/><\/p>\n<p>Question 6.<br \/>\nUW<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41297\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-9.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 9\" width=\"155\" height=\"105\" \/><\/p>\n<p>Answer:<br \/>\nUW = 55<\/p>\n<p>Explanation:<br \/>\nUV = WU<br \/>\n9x + 1 = 7x + 13<br \/>\n9x &#8211; 7x = 13 &#8211; 1<br \/>\n2x = 12<br \/>\nx = 6<br \/>\nUW = 7(6) + 13 = 42 + 13 = 55<\/p>\n<p>In Exercises 7-10. tell whether the information in the diagram allows you to conclude that point P lies on the perpendicular bisector of \\(\\overline{L M}\\). Explain your reasoning.<\/p>\n<p>Question 7.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41296\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-10.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 10\" width=\"126\" height=\"108\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48933\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-7.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 7\" width=\"583\" height=\"138\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-7.png 583w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-7-300x71.png 300w\" sizes=\"(max-width: 583px) 100vw, 583px\" \/><\/p>\n<p>Question 8.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41295\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-11.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 11\" width=\"121\" height=\"123\" \/><\/p>\n<p>Answer:<br \/>\nPN \u22a5 LM. So, LN &#8211; MN<br \/>\nTherefore, point P lies on the LM.<\/p>\n<p>Question 9.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41294\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-12.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 12\" width=\"141\" height=\"136\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48934\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-9.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 9\" width=\"409\" height=\"37\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-9.png 409w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-9-300x27.png 300w\" sizes=\"(max-width: 409px) 100vw, 409px\" \/><\/p>\n<p>Question 10.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41293\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-13.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 13\" width=\"172\" height=\"123\" \/><\/p>\n<p>Answer:<br \/>\nYes.<\/p>\n<p>In Exercises 11-14. find the indicated measure. Explain your reasoning.<\/p>\n<p>Question 11.<br \/>\nm\u2220ABD<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41292\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-14.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 14\" width=\"160\" height=\"115\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48935\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-11.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 11\" width=\"580\" height=\"89\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-11.png 580w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-11-300x46.png 300w\" sizes=\"(max-width: 580px) 100vw, 580px\" \/><\/p>\n<p>Question 12.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41291\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-15.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 15\" width=\"114\" height=\"108\" \/><\/p>\n<p>Answer:<br \/>\nPS = RS<br \/>\nPS = 12<\/p>\n<p>Question 13.<br \/>\nm\u2220KJL<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41290\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-16.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 16\" width=\"139\" height=\"123\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48936\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-13.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 13\" width=\"565\" height=\"208\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-13.png 565w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-13-300x110.png 300w\" sizes=\"(max-width: 565px) 100vw, 565px\" \/><\/p>\n<p>Question 14.<br \/>\nFG<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41289\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-17.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 17\" width=\"139\" height=\"116\" \/><\/p>\n<p>Answer:<br \/>\nFG = 16<\/p>\n<p>Explanation:<br \/>\nFG = GH<br \/>\nx + 11 = 3x + 1<br \/>\n3x &#8211; x = 11 &#8211; 1<br \/>\n2x = 10<br \/>\nx = 5<br \/>\nFG = 5 + 11 = 16<\/p>\n<p>In Exercises 15 and 16, tell whether the information in the diagram allows you to conclude that \\(\\vec{E}\\)H bisects \u2220FEG. Explain your reasoning.<\/p>\n<p>Question 15.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41288\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-18.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 18\" width=\"121\" height=\"107\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48937\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-15.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 15\" width=\"547\" height=\"65\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-15.png 547w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-15-300x36.png 300w\" sizes=\"(max-width: 547px) 100vw, 547px\" \/><\/p>\n<p>Question 16.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41287\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-19.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 19\" width=\"122\" height=\"107\" \/><\/p>\n<p>Answer:<br \/>\nYes, because H is equidistant from EF and EG, EH bisects \u2220FEG by the Angle Bisector Theorem.<\/p>\n<p>In Exercises 17 and 18, tell whether the information in the diagram allows you to conclude that DB = DC. Explain your reasoning.<\/p>\n<p>Question 17.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41286\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-20.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 20\" width=\"129\" height=\"95\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48938\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-17.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 17\" width=\"586\" height=\"65\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-17.png 586w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-17-300x33.png 300w\" sizes=\"(max-width: 586px) 100vw, 586px\" \/><\/p>\n<p>Question 18.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41285\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-21.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 21\" width=\"125\" height=\"90\" \/><\/p>\n<p>Answer:<br \/>\nYes, because AD is perpendicular to BD and DC. using perpendicular bisector theorem DB = DC.<\/p>\n<p>In Exercises 19-22, write an equation of the perpendicular bisector of the segment with the given endpoints.<\/p>\n<p>Question 19.<br \/>\nM(1, 5), N(7, &#8211; 1)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48939\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-19.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 19\" width=\"570\" height=\"356\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-19.png 570w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-19-300x187.png 300w\" sizes=\"(max-width: 570px) 100vw, 570px\" \/><\/p>\n<p>Question 20.<br \/>\nQ(- 2, 0), R(6, 12)<\/p>\n<p>Answer:<br \/>\nAn equation of the perpendicular bisector is y = &#8211;\\(\\frac { 3 }{ 7 } \\)x + \\(\\frac { 36 }{ 7 } \\)<\/p>\n<p>Explanation:<br \/>\nSlope of QR = \\(\\frac { 12 + 2 }{ 6 &#8211; 0 } \\) = \\(\\frac { 7 }{ 3 } \\)<br \/>\nThe slope of the perpendicular line m = &#8211;\\(\\frac { 3 }{ 7 } \\)<br \/>\nMidpoint of QR = (\\(\\frac { -2 + 6 }{ 2 } \\), \\(\\frac { 0 + 12 }{ 2 } \\)<br \/>\n= (-2, 6)<br \/>\ny = mx + b<br \/>\n6 = &#8211;\\(\\frac { 3 }{ 7 } \\)(-2) + b<br \/>\n6 = \\(\\frac { 6 }{ 7 } \\) + b<br \/>\nb = \\(\\frac { 36 }{ 7 } \\)<br \/>\nAn equation of the perpendicular bisector is y = &#8211;\\(\\frac { 3 }{ 7 } \\)x + \\(\\frac { 36 }{ 7 } \\)<\/p>\n<p>Question 21.<br \/>\nU(- 3, 4), V(9, 8)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48940\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-21.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 21\" width=\"517\" height=\"389\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-21.png 517w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-21-300x226.png 300w\" sizes=\"(max-width: 517px) 100vw, 517px\" \/><\/p>\n<p>Question 22.<br \/>\nY( 10, &#8211; 7), Z(- 4, 1)<\/p>\n<p>Answer:<br \/>\nAn equation of the perpendicular bisector is y = \\(\\frac { 7 }{ 4 } \\)x &#8211; \\(\\frac { 33 }{ 4 } \\)<\/p>\n<p>Explanation:<br \/>\nSlope of YZ = \\(\\frac { 1 + 7 }{ -4 &#8211; 10 } \\) = \\(\\frac { -4 }{ 7 } \\)<br \/>\nThe slope of the perpendicular line m = \\(\\frac { 7 }{ 4 } \\)<br \/>\nMidpoint of YZ = (\\(\\frac { 10 &#8211; 4 }{ 2 } \\), \\(\\frac { -7 + 1 }{ 2 } \\))<br \/>\n= (3, -3)<br \/>\ny = mx + b<br \/>\n-3 = \\(\\frac { 7 }{ 4 } \\)(3) + b<br \/>\nb = -3 &#8211; \\(\\frac { 21 }{ 4 } \\) = \\(\\frac { -12 &#8211; 21 }{ 4 } \\) = \\(\\frac { -33 }{ 4 } \\)<br \/>\nAn equation of the perpendicular bisector is y = \\(\\frac { 7 }{ 4 } \\)x &#8211; \\(\\frac { 33 }{ 4 } \\)<\/p>\n<p><strong>ERROR ANALYSIS<\/strong><br \/>\nIn Exercises 23 and 24, describe and correct the error in the student\u2019s reasoning.<\/p>\n<p>Question 23.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41284\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-22.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 22\" width=\"332\" height=\"102\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-22.png 332w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-22-300x92.png 300w\" sizes=\"(max-width: 332px) 100vw, 332px\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48941\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-23.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 23\" width=\"576\" height=\"117\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-23.png 576w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-23-300x61.png 300w\" sizes=\"(max-width: 576px) 100vw, 576px\" \/><\/p>\n<p>Question 24.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41283\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-23.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 23\" width=\"327\" height=\"128\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-23.png 327w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-23-300x117.png 300w\" sizes=\"(max-width: 327px) 100vw, 327px\" \/><\/p>\n<p>Answer:<br \/>\nCP is not perpendicular to BP. So, x = 5 is wrong.<\/p>\n<p>Question 25.<br \/>\n<strong>MODELING MATHEMATICS<\/strong><br \/>\nIn the photo, the road is perpendicular to the support beam and \\(\\overline{A B} \\cong \\overline{C B}\\). Which theorem allows you to conclude that \\(\\overline{A D} \\cong \\overline{C D}\\)?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41282\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-24.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 24\" width=\"263\" height=\"147\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48942\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-25.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 25\" width=\"578\" height=\"59\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-25.png 578w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-25-300x31.png 300w\" sizes=\"(max-width: 578px) 100vw, 578px\" \/><\/p>\n<p>Question 26.<br \/>\n<strong>MODELING WITH MATHEMATICS<\/strong><br \/>\nThe diagram shows the position of the goalie and the puck during a hockey game. The goalie is at point G. and the puck is at point P.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41281\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-25.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 25\" width=\"194\" height=\"188\" \/><br \/>\na. What should be the relationship between \\(\\vec{P}\\)G and \u2220APB to give the goalie equal distances to travel on each side of \\(\\vec{P}\\)G?<\/p>\n<p>Answer:<br \/>\nPG should be the perpendicular bisector of \u2220APB.<\/p>\n<p>b. How does m\u2220APB change as the puck gets closer to the goal? Does this change make it easier or more difficult for the goalie to defend the goal? Explain your reasoning.<br \/>\nAnswer:<\/p>\n<p>Question 27.<br \/>\n<strong>CONSTRUCTION<\/strong><br \/>\nUse a compass and straightedge to construct a copy of \\(\\overline{X Y}\\). Construct a perpendicular bisector and plot a point Z on the bisector so that the distance between point Z and \\(\\overline{X Y}\\) is 3 centimeters. Measure \\(\\overline{X Z}\\) and \\(\\overline{Y Z}\\). Which theorem does this construction demonstrate?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41280\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-26.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 26\" width=\"262\" height=\"33\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48943\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-27.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 27\" width=\"582\" height=\"458\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-27.png 582w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-27-300x236.png 300w\" sizes=\"(max-width: 582px) 100vw, 582px\" \/><\/p>\n<p>Question 28.<br \/>\n<strong>WRITING<\/strong><br \/>\nExplain how the Converse of the Perpendicular Bisector Theorem (Theorem 6.2) is related to the construction of a perpendicular bisector.<\/p>\n<p>Answer:<br \/>\nThe converse of the perpendicular bisector theorem states if a point is equidistant from both the endpoints of the line segments in the same plane, then that point is on the perpendicular bisector of the line segment.<\/p>\n<p>Question 29.<br \/>\n<strong>REASONING<\/strong><br \/>\nWhat is the value of x in the diagram?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41279\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-27.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 27\" width=\"138\" height=\"96\" \/><br \/>\n(A) 13<br \/>\n(B) 18<br \/>\n(C) 33<br \/>\n(D) not enough information<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48944\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-29.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 29\" width=\"192\" height=\"101\" \/><\/p>\n<p>Question 30.<br \/>\n<strong>REASONING<\/strong><br \/>\nWhich point lies on the perpendicular bisector of the segment with endpoints M(7, 5) and m(- 1, 5)?<br \/>\n(A) (2, 0)<br \/>\n(B) (3, 9)<br \/>\n(C) (4, 1)<br \/>\n(D) (1, 3)<\/p>\n<p>Answer:<br \/>\n(B) (3, 9)<\/p>\n<p>Explanation:<br \/>\nThe distance from M to (3, 9) is = \u221a(7 &#8211; 3)\u00b2 + (5 &#8211; 9)\u00b2 = \u221a16 + 16 = \u221a32<br \/>\nThe distance from m to (3, 9) = \u221a(-1 &#8211; 3)\u00b2 + (5 &#8211; 9)\u00b2 = \u221a16 + 16 = \u221a32<\/p>\n<p>Question 31.<br \/>\n<strong>MAKING AN ARGUMENT<\/strong><br \/>\nYour friend says it is impossible for an angle bisector of a triangle to be the same line as the perpendicular bisector of the opposite side. Is your friend correct? Explain your reasoning.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48945\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-31.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 31\" width=\"577\" height=\"112\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-31.png 577w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-31-300x58.png 300w\" sizes=\"(max-width: 577px) 100vw, 577px\" \/><\/p>\n<p>Question 32.<br \/>\n<strong>PROVING A THEOREM<\/strong><br \/>\nProve the Converse of the Perpendicular Bisector Theorem (Thm. 6.2). (Hint: Construct a line through point C perpendicular to \\(\\overline{A B}\\) at point P.)<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41278\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-28.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 28\" width=\"154\" height=\"94\" \/><br \/>\nGiven CA = CB<br \/>\nProve Point C lies on the perpendicular bisector of \\(\\overline{A B}\\).<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-105368\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-geometry-answers-chapter-6-relationships-within-triangles-1-2.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 1\" width=\"281\" height=\"371\" \/><br \/>\nCompare \u25b3ACP, \u25b3CPB<br \/>\nAP = BP<br \/>\nCP = CP<br \/>\n\u2220APC = \u2220BPC = 90\u00b0<br \/>\nSo, \u25b3ACP \u2245 \u25b3CPB using SAS congruence theorem<br \/>\nSo, CA = CB<\/p>\n<p>Question 33.<br \/>\n<strong>PROVING A THEOREM<\/strong><br \/>\nUse a congruence theorem to prove each theorem.<br \/>\na. Angle Bisector Theorem (Thin. 6.3)<br \/>\nb. Converse of the Angle Bisector Theorem (Thm. 6.4)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48946\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-33.1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 33.1\" width=\"774\" height=\"569\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-33.1.png 774w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-33.1-300x221.png 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-33.1-768x565.png 768w\" sizes=\"(max-width: 774px) 100vw, 774px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48947\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-33.2.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 33.2\" width=\"781\" height=\"625\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-33.2.png 781w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-33.2-300x240.png 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-33.2-768x615.png 768w\" sizes=\"(max-width: 781px) 100vw, 781px\" \/><\/p>\n<p>Question 34.<br \/>\n<strong>HOW DO YOU SEE IT?<\/strong><br \/>\nThe figure shows a map of a city. The city is arranged so each block north to south is the same length and each block east to west is the same length.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41277\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-29.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 29\" width=\"357\" height=\"170\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-29.png 357w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-29-300x143.png 300w\" sizes=\"(max-width: 357px) 100vw, 357px\" \/><br \/>\na. Which school is approximately equidistant from both hospitals? Explain your reasoning.<br \/>\nAnswer:<\/p>\n<p>b. Is the museum approximately equidistant from Wilson School and Roosevelt School? Explain your reasoning.<br \/>\nAnswer:<\/p>\n<p>Question 35.<br \/>\n<strong>MATHEMATICAL CONNECTIONS<\/strong><br \/>\nWrite an equation whose graph consists of all the points in the given quadrants that are equidistant from the x- and y-axes.<br \/>\na. I and III<br \/>\nb. II and IV<br \/>\nc. I and II<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48948\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-35.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 35\" width=\"360\" height=\"38\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-35.png 360w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-35-300x32.png 300w\" sizes=\"(max-width: 360px) 100vw, 360px\" \/><\/p>\n<p>Question 36.<br \/>\n<strong>THOUGHT PROVOKING<\/strong><br \/>\nThe postulates and theorems in this book represent Euclidean geometry. In spherical geometry, all points are on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, is it possible for two lines to be perpendicular but not bisect each other? Explain your reasoning.<br \/>\nAnswer:<\/p>\n<p>Question 37.<br \/>\n<strong>PROOF<\/strong><br \/>\nUse the information in the diagram to prove that \\(\\overline{A B} \\cong \\overline{C B}\\) if and onI if points D, E, and B are collinear.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41276\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-30.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 30\" width=\"197\" height=\"118\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48949\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-37.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 37\" width=\"585\" height=\"497\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-37.png 585w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-37-300x255.png 300w\" sizes=\"(max-width: 585px) 100vw, 585px\" \/><\/p>\n<p>Question 38.<br \/>\n<strong>PROOF<\/strong><br \/>\nprove the statement in parts (a) &#8211; (c)<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41275\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-31.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 31\" width=\"193\" height=\"127\" \/><br \/>\nGiven Plane P is a perpendicular bisector of \\(\\overline{X Z}\\) at point Y.<br \/>\nProve<br \/>\na. \\(\\overline{X W} \\cong \\overline{Z W}\\)<br \/>\nb. \\(\\overline{X V} \\cong \\overline{Z V}\\)<br \/>\nc. \u2220VXW \u2245 \u2220VZW<\/p>\n<p>Answer:<br \/>\na. By using the perpendicular bisector theorem \\(\\overline{X W} \\cong \\overline{Z W}\\)<br \/>\nb. By using the perpendicular bisector theorem \\(\\overline{X V} \\cong \\overline{Z V}\\)<\/p>\n<p>Maintaining Mathematical Proficiency<\/p>\n<p>Classify the triangle by its sides.<\/p>\n<p>Question 39.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41274\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-32.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 32\" width=\"81\" height=\"83\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48950\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-39.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 39\" width=\"555\" height=\"36\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-39.png 555w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-39-300x19.png 300w\" sizes=\"(max-width: 555px) 100vw, 555px\" \/><\/p>\n<p>Question 40.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41273\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-33.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 33\" width=\"117\" height=\"59\" \/><\/p>\n<p>Answer:<br \/>\nThe trinagle is scalene as all the sides have different lengths.<\/p>\n<p>Question 41.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41272\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-34.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 34\" width=\"86\" height=\"85\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48951\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-41.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 41\" width=\"530\" height=\"34\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-41.png 530w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-41-300x19.png 300w\" sizes=\"(max-width: 530px) 100vw, 530px\" \/><\/p>\n<p>Classify the triangles by its angles.<\/p>\n<p>Question 42.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41271\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-35.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 35\" width=\"135\" height=\"88\" \/><br \/>\nAnswer:<br \/>\nThe triangle is scalene as all angles are different.<\/p>\n<p>Question 43.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41270\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-36.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 36\" width=\"139\" height=\"70\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48952\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-43.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.1 Question 43\" width=\"558\" height=\"34\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-43.png 558w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.1-Question-43-300x18.png 300w\" sizes=\"(max-width: 558px) 100vw, 558px\" \/><\/p>\n<p>Question 44.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41269\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-37.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 37\" width=\"157\" height=\"64\" \/><\/p>\n<p>Answer:<br \/>\nThe trinagle is an obtuse triangle.<\/p>\n<h3><a id=\"6.2_Bisectors_of_Triangles\"><\/a>6.2 Bisectors of Triangles<\/h3>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-41728\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-38.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 38\" width=\"462\" height=\"240\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-38.png 462w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-38-300x156.png 300w\" sizes=\"(max-width: 462px) 100vw, 462px\" \/><\/p>\n<p><strong>Exploration 1<\/strong><\/p>\n<p>Properties of the Perpendicular Bisectors of a Triangle<\/p>\n<p>Work with a partner: Use dynamic geometry software. Draw any \u2206ABC.<br \/>\na. Construct the perpendicular bisectors of all three sides of \u2206ABC. Then drag the vertices to change \u2206ABC. \u2018What do you notice about the perpendicular bisectors?<br \/>\nAnswer:<\/p>\n<p>b. Label a point D at the intersection of the perpendicular bisectors.<br \/>\nAnswer:<\/p>\n<p>c. Draw the circle with center D through vertex A of \u2206ABC. Then drag the vertices to change \u2206ABC. What do you notice?<br \/>\nAnswer:<\/p>\n<p><strong>Exploration 2<\/strong><\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-41726\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-39.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 39\" width=\"523\" height=\"240\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-39.png 523w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-39-300x138.png 300w\" sizes=\"(max-width: 523px) 100vw, 523px\" \/><\/p>\n<p>Properties of the Angle Bisectors of a Triangle<\/p>\n<p>Work with a partner. Use dynamic geometry software. Draw any \u2206ABC.<\/p>\n<p>a. Construct the angle bisectors of all three angles of \u2206ABC, Then drag the vertices to change \u2206ABC. What do you notice about the angle bisectors?<br \/>\nAnswer:<\/p>\n<p>b. Label a point D at the intersection of the angle bisectors.<br \/>\nAnswer:<\/p>\n<p>c. Find the distance between D and \\(\\overline{A B}\\). Draw the circle with center D and this distance as a radius. Then drag the vertices to change \u2206ABC. What do you notice?<br \/>\n<strong>LOOKING FOR STRUCTURE<\/strong><br \/>\nTo be proficient in math, you need to see complicated things as single objects or as being composed of several objects.<br \/>\nAnswer:<\/p>\n<p>Communicate Your Answer<\/p>\n<p>Question 3.<br \/>\nWhat conjectures can you make about the perpendicular bisectors and the angle bisectors of a triangle?<br \/>\nAnswer:<\/p>\n<h3><a id=\"Lesson_6.2_Bisectors_of_Triangles\"><\/a>Lesson 6.2 Bisectors of Triangles<\/h3>\n<p><strong>Monitoring Progress<\/strong><\/p>\n<p>Question 1.<br \/>\nThree snack carts sell hot pretzels horn points A, B, and E. What is the location of the pretzel distributor if it is equidistant from the three carts? Sketch the triangle and show the location.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41725\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-40.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 40\" width=\"196\" height=\"121\" \/><\/p>\n<p>Answer:<br \/>\nThe location is the circumcenter<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-105373\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-geometry-answers-chapter-6-relationships-within-triangles-6.2-1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 1\" width=\"318\" height=\"275\" \/><\/p>\n<p>Find the coordinates of the circumcenter of the triangle with the given vertices.<\/p>\n<p>Question 2.<br \/>\nR(- 2, 5), S(- 6, 5), T(- 2, &#8211; 1)<\/p>\n<p>Answer:<br \/>\nThe circumcenter is (4, 7.33)<\/p>\n<p>Explanation:<br \/>\nRecall that the circumcentre of a triangle is equidistant from the vertices of a triangle. Let R(- 2, 5), S(- 6, 5), T(- 2, &#8211; 1) be the vertices of the given triangle and let P(x,y) be the circumcentre of this triangle. Then<br \/>\nPR = PS = PT<br \/>\nPR\u00b2 = PS\u00b2 = PT\u00b2<br \/>\nPR\u00b2 = PS\u00b2<br \/>\n(x + 2)\u00b2 + (y &#8211; 5)\u00b2 = (x + 6)\u00b2 + (y &#8211; 5)\u00b2<br \/>\nx\u00b2 + 4x + 4 + y\u00b2 &#8211; 10x + 25 = x\u00b2 + 12x + 36 + y\u00b2 -10y + 25<br \/>\n4x + 4 = 12x + 36<br \/>\n12x &#8211; 4x = 36 &#8211; 4<br \/>\n8x = 32<br \/>\nx = 4<br \/>\nPS\u00b2 = PT\u00b2<br \/>\n(x + 6)\u00b2 + (y &#8211; 5)\u00b2 = (x + 2)\u00b2 + (y + 1)\u00b2<br \/>\nx\u00b2 + 12x + 36 + y\u00b2 -10y + 25 = x\u00b2 + 4x + 4 + y\u00b2 + 2y + 1<br \/>\n12x &#8211; 4x &#8211; 10y &#8211; 2y + 61 = 5<br \/>\n8x &#8211; 12y = -56<br \/>\n8(4) &#8211; 12y = -56<br \/>\n32 &#8211; 12y = -56<br \/>\n-12y = -56 &#8211; 32<br \/>\n-12y = -88<br \/>\ny = 7.33<br \/>\nThe circumcenter is (4, 7.33)<\/p>\n<p>Question 3.<br \/>\nW(- 1, 4), X(1, 4), Y( 1, &#8211; 6)<\/p>\n<p>Answer:<br \/>\nThe circumcenter is (0.25, -1)<\/p>\n<p>Explanation:<br \/>\nRecall that the circumcentre of a triangle is equidistant from the vertices of a triangle. Let W(- 1, 4), X(1, 4), Y( 1, &#8211; 6) be the vertices of the given triangle and let P(x,y) be the circumcentre of this triangle. Then<br \/>\nPW = PX = PY<br \/>\nPW\u00b2 = PY\u00b2 = PX\u00b2<br \/>\nPX\u00b2 = PY\u00b2<br \/>\n(x &#8211; 1)\u00b2 + (y &#8211; 4)\u00b2 = (x &#8211; 1)\u00b2 + (y + 6)\u00b2<br \/>\nx\u00b2 -2x + 1 + y\u00b2 &#8211; 8y + 16 = x\u00b2 -2x + 1 + y\u00b2 + 12y + 36<br \/>\n-8y &#8211; 12y = 36 &#8211; 16<br \/>\n-20y = 20<br \/>\ny = -1<br \/>\nPW\u00b2 = PX\u00b2<br \/>\n(x + 1)\u00b2 + (y &#8211; 4)\u00b2 = (x &#8211; 1)\u00b2 + (y &#8211; 4)\u00b2<br \/>\nx\u00b2 + 2x + 4 + y\u00b2 &#8211; 8y + 16 = x\u00b2 &#8211; 2x + 1 + y\u00b2 &#8211; 8y + 16<br \/>\n4x = 1<br \/>\nx = 0.25<br \/>\nThe circumcenter is (0.25, -1)<\/p>\n<p>Question 4.<br \/>\nIn the figure shown, QM = 3x + 8 and QN = 7x + 2. Find QP.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41724\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-41.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 41\" width=\"203\" height=\"118\" \/><\/p>\n<p>Answer:<br \/>\nQP = \\(\\frac { 23 }{ 2 } \\)<\/p>\n<p>Explanation:<br \/>\nQM = QN<br \/>\n3x + 8 = 7x + 2<br \/>\n7x &#8211; 3x = 8 &#8211; 2<br \/>\n4x = 6<br \/>\nx = \\(\\frac { 3 }{ 2 } \\)<br \/>\nQP = QN<br \/>\n= 7(\\(\\frac { 3 }{ 2 } \\)) + 2<br \/>\n= \\(\\frac { 23 }{ 2 } \\)<\/p>\n<p>Question 5.<br \/>\nDraw a sketch to show the location L of the lamppost in Example 4.<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-105394\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-geometry-answers-chapter-6-relationships-within-triangles-6.2-2.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 2\" width=\"329\" height=\"238\" \/><\/p>\n<h3><a id=\"Exercise_6.2_Bisectors_of_Triangles\"><\/a>Exercise 6.2 Bisectors of Triangles<\/h3>\n<p>Vocabulary and Core Concept Check<\/p>\n<p>Question 1.<br \/>\n<strong>VOCABULARY<\/strong><br \/>\nWhen three or more lines, rays, or segments intersect in the same Point. they are called _____________ lines, rays, or segments.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48953\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 1\" width=\"576\" height=\"61\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-1.png 576w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-1-300x32.png 300w\" sizes=\"(max-width: 576px) 100vw, 576px\" \/><\/p>\n<p>Question 2.<br \/>\n<strong>WHICH ONE DOESNT BELONG?<\/strong><br \/>\nWhich triangle does not belong with the other three? Explain your reasoning.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41723\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-42.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 42\" width=\"706\" height=\"137\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-42.png 706w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-42-300x58.png 300w\" sizes=\"(max-width: 706px) 100vw, 706px\" \/><\/p>\n<p>Answer:<br \/>\nThe third triangle does not belong with the other three. Because the point P in the remaining triangles is the circumcenter. But P is not circumcenter in the third triangle.<\/p>\n<p>Monitoring Progress and Modeling with Mathematics<\/p>\n<p>In Exercises 3 and 4, the perpendicular bisectors of \u2206ABC intersect at point G and are shown in blue. Find the indicated measure.<\/p>\n<p>Question 3.<br \/>\nFind BG<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41722\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-43.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 43\" width=\"169\" height=\"132\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48954\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-3.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 3\" width=\"543\" height=\"58\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-3.png 543w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-3-300x32.png 300w\" sizes=\"(max-width: 543px) 100vw, 543px\" \/><\/p>\n<p>Question 4.<br \/>\nFind GA<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41721\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-44.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 44\" width=\"202\" height=\"120\" \/><\/p>\n<p>Answer:<br \/>\nBecause G is the circumcenter of \u2206ABC, AG = BG = CG<br \/>\nAG = BG = 11<br \/>\nSo, AG = 11<\/p>\n<p>In Exercises 5 and 6, the angle bisectors of \u2206XYZ intersect at point P and are shown in red. Find the indicated measure.<\/p>\n<p>Question 5.<br \/>\nFind PB.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41720\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-45.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 45\" width=\"193\" height=\"133\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48955\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-5.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 5\" width=\"486\" height=\"61\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-5.png 486w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-5-300x38.png 300w\" sizes=\"(max-width: 486px) 100vw, 486px\" \/><\/p>\n<p>Question 6.<br \/>\nFind HP.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41719\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-46.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 46\" width=\"204\" height=\"125\" \/><\/p>\n<p>Answer:<br \/>\nBecause P is the incenter of \u2206XYZ, PH = PF = PK<br \/>\nTherefore, PK = 15<br \/>\nHP = 15<\/p>\n<p>In Exercises 7-10. find the coordinates of the circumcenter of the triangle with the given vertice<\/p>\n<p>Question 7.<br \/>\nA(2, 6), B(8, 6), C(8, 10)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48956\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-7.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 7\" width=\"578\" height=\"549\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-7.png 578w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-7-300x285.png 300w\" sizes=\"(max-width: 578px) 100vw, 578px\" \/><\/p>\n<p>Question 8.<br \/>\nD(- 7, &#8211; 1), E(- 1, &#8211; 1), F(- 7, &#8211; 9)<\/p>\n<p>Answer:<br \/>\nThe circumcenter is (-4, -5)<\/p>\n<p>Explanation:<br \/>\nRecall that the circumcentre of a triangle is equidistant from the vertices of a triangle. Let D(- 7, &#8211; 1), E(- 1, &#8211; 1), F(- 7, &#8211; 9) be the vertices of the given triangle and let P(x,y) be the circumcentre of this triangle. Then<br \/>\nPD = PE = PF<br \/>\nPD\u00b2 = PE\u00b2 = PF\u00b2<br \/>\nPD\u00b2 = PE\u00b2<br \/>\n(x + 7)\u00b2 + (y + 1)\u00b2 = (x + 1)\u00b2 + (y + 1)\u00b2<br \/>\nx\u00b2 + 14x + 49 + y\u00b2 + 2y +1 = x\u00b2 + 2x + 1 + y\u00b2 + 2y + 1<br \/>\n14x &#8211; 2x = 1 &#8211; 49<br \/>\n12x = -48<br \/>\nx = -4<br \/>\nPD\u00b2 = PF\u00b2<br \/>\n(x + 7)\u00b2 + (y + 1)\u00b2 = (x + 7)\u00b2 + (y + 9)\u00b2<br \/>\nx\u00b2 + 14x + 49 + y\u00b2 + 2y + 1 = x\u00b2 + 14x + 49 + y\u00b2 + 18y + 81<br \/>\n18y &#8211; 2y = 1 &#8211; 81<br \/>\n16y = -80<br \/>\ny = -5<br \/>\nThe circumcenter is (-4, -5)<\/p>\n<p>Question 9.<br \/>\nH(- 10, 7), J(- 6, 3), K(- 2, 3)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48957\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-9.1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 9.1\" width=\"552\" height=\"526\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-9.1.png 552w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-9.1-300x286.png 300w\" sizes=\"(max-width: 552px) 100vw, 552px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48958\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-9.2.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 9.2\" width=\"736\" height=\"563\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-9.2.png 736w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-9.2-300x229.png 300w\" sizes=\"(max-width: 736px) 100vw, 736px\" \/><\/p>\n<p>Question 10.<br \/>\nL(3, &#8211; 6), M(5, &#8211; 3) , N (8, &#8211; 6)<\/p>\n<p>Answer:<br \/>\nThe circumcenter is (5.5, -5.5)<\/p>\n<p>Explanation:<br \/>\nRecall that the circumcentre of a triangle is equidistant from the vertices of a triangle. Let L(3, &#8211; 6), M(5, &#8211; 3) , N (8, &#8211; 6) be the vertices of the given triangle and let P(x,y) be the circumcentre of this triangle. Then<br \/>\nPL = PM = PN<br \/>\nPL\u00b2 = PM\u00b2 = PN\u00b2<br \/>\nPL\u00b2 = PN\u00b2<br \/>\n(x &#8211; 3)\u00b2 + (y + 6)\u00b2 = (x &#8211; 8)\u00b2 + (y + 6)\u00b2<br \/>\nx\u00b2 &#8211; 6x + 9 + y\u00b2 + 12y + 36 = x\u00b2 -16x + 64 + y\u00b2 + 12y + 36<br \/>\n-16x + 6x = 9 &#8211; 64<br \/>\n-10x = -55<br \/>\nx = 5.5<br \/>\nPL\u00b2 = PM\u00b2<br \/>\n(x &#8211; 3)\u00b2 + (y + 6)\u00b2 = (x &#8211; 5)\u00b2 + (y + 3)\u00b2<br \/>\nx\u00b2 &#8211; 6x + 9 + y\u00b2 + 12y + 36 = x\u00b2 &#8211; 10x + 25 + y\u00b2 + 6y + 9<br \/>\n-6x + 10x + 45 = 6y &#8211; 12y + 34<br \/>\n4x = -6y -11<br \/>\n4(5.5) = -6y &#8211; 11<br \/>\n22 + 11 = -6y<br \/>\n33 = -6y<br \/>\ny = -5.5<br \/>\nThe circumcenter is (5.5, -5.5)<\/p>\n<p>In Exercises 11-14, N is the incenter of \u2206ABC. Use the given information to find the indicated measure.<\/p>\n<p>Question 11.<br \/>\nND = 6x &#8211; 2<br \/>\nNE = 3x + 7<br \/>\nFind NF.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41718\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-47.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 47\" width=\"183\" height=\"129\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48959\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-11.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 11\" width=\"567\" height=\"249\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-11.png 567w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-11-300x132.png 300w\" sizes=\"(max-width: 567px) 100vw, 567px\" \/><\/p>\n<p>Question 12.<br \/>\nNG = x + 3<br \/>\nNH = 2x &#8211; 3<br \/>\nFind NJ.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41717\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-48.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 48\" width=\"158\" height=\"120\" \/><\/p>\n<p>Answer:<br \/>\nNJ = 9<\/p>\n<p>Explanation:<br \/>\nNG = NH = NJ<br \/>\nx + 3 = 2x &#8211; 3<br \/>\n2x &#8211; x = 3 + 3<br \/>\nx = 6<br \/>\nBy Incenter theorem, NG = NH = NJ<br \/>\nNJ = 6 + 3 = 9<\/p>\n<p>Question 13.<br \/>\nNK = 2x &#8211; 2<br \/>\nNL = &#8211; x + 10<br \/>\nFind NM<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41716\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-49.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 49\" width=\"133\" height=\"125\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48960\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-13.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 13\" width=\"563\" height=\"248\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-13.png 563w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-13-300x132.png 300w\" sizes=\"(max-width: 563px) 100vw, 563px\" \/><\/p>\n<p>Question 14.<br \/>\nNQ = 2x<br \/>\nNR = 3x &#8211; 2<br \/>\nFind NS.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41715\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-50.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 50\" width=\"172\" height=\"125\" \/><\/p>\n<p>Answer:<br \/>\nNS = 4<\/p>\n<p>Explanation:<br \/>\nNQ = NR<br \/>\n2x = 3x &#8211; 2<br \/>\n3x &#8211; 2x = 2<br \/>\nx = 2<br \/>\nNQ = 2 (2) = 4<br \/>\nBy Incenter theorem NS = NR = NQ<br \/>\nSo, NS = 4<\/p>\n<p>Question 15.<br \/>\nP is the circumcenter of \u2206XYZ. Use the given information to find PZ.<br \/>\nPX = 3x + 2<br \/>\nPY = 4x &#8211; 8<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41714\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-51.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 51\" width=\"177\" height=\"126\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48961\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-15.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 15\" width=\"555\" height=\"239\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-15.png 555w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-15-300x129.png 300w\" sizes=\"(max-width: 555px) 100vw, 555px\" \/><\/p>\n<p>Question 16.<br \/>\nP is the circumcenter of \u2206XYZ. Use the given information to find PY.<br \/>\nPX = 4x + 3<br \/>\nPZ = 6x &#8211; 11<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41713\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-52.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 52\" width=\"160\" height=\"103\" \/><\/p>\n<p>Answer:<br \/>\nPY = 31<\/p>\n<p>Explanation:<br \/>\nPX = PZ<br \/>\n4x + 3 = 6x &#8211; 11<br \/>\n6x &#8211; 4x = 3 + 11<br \/>\n2x = 14<br \/>\nx = 7<br \/>\nPZ = 6(7) &#8211; 11 = 42 &#8211; 11 = 31<br \/>\nBy the incenter theorem, PX = PZ = PY<br \/>\nPY = 31<\/p>\n<p><strong>CONSTRUCTION<\/strong><br \/>\nIn Exercises 17-20. draw a triangle of the given type. Find the circumcenter. Then construct the circumscribed circle.<\/p>\n<p>Question 17.<br \/>\nright<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48962\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-17.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 17\" width=\"539\" height=\"278\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-17.png 539w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-17-300x155.png 300w\" sizes=\"(max-width: 539px) 100vw, 539px\" \/><\/p>\n<p>Question 18.<br \/>\nobtuse<\/p>\n<p>Answer:<\/p>\n<p>Question 19.<br \/>\nacute isosceles<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48963\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-19.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 19\" width=\"536\" height=\"278\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-19.png 536w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-19-300x156.png 300w\" sizes=\"(max-width: 536px) 100vw, 536px\" \/><\/p>\n<p>Question 20.<br \/>\nequilateral<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-105498\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-geometry-answers-chapter-6-relationships-within-triangles-6.2-3.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 3\" width=\"348\" height=\"335\" \/><\/p>\n<p><strong>CONSTRUCTION<\/strong><br \/>\nIn Exercises 21-24, copy the triangle with the given angle measures. Find the incenter. Then construct the inscribed circle.<\/p>\n<p>Question 21.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41712\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-53.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 53\" width=\"150\" height=\"128\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48964\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-21.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 21\" width=\"578\" height=\"422\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-21.png 578w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-21-300x219.png 300w\" sizes=\"(max-width: 578px) 100vw, 578px\" \/><\/p>\n<p>Question 22.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41711\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-54.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 54\" width=\"199\" height=\"112\" \/><\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-105506\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-geometry-answers-chapter-6-relationships-within-triangles-6.2-4.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 4\" width=\"334\" height=\"227\" \/><\/p>\n<p>Question 23.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41710\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-55.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 55\" width=\"260\" height=\"104\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48965\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-23.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 23\" width=\"575\" height=\"428\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-23.png 575w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-23-300x223.png 300w\" sizes=\"(max-width: 575px) 100vw, 575px\" \/><\/p>\n<p>Question 24.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41709\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-56.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 56\" width=\"144\" height=\"131\" \/><\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-105513\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-geometry-answers-chapter-6-relationships-within-triangles-6.2-5.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 5\" width=\"327\" height=\"309\" \/><\/p>\n<p><strong>ERROR ANALYSIS<\/strong><br \/>\nIn Exercises 25 and 26. describe and correct the error in identifying equal distances inside the triangle.<\/p>\n<p>Question 25.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41708\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-57.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 57\" width=\"330\" height=\"121\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-57.png 330w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-57-300x110.png 300w\" sizes=\"(max-width: 330px) 100vw, 330px\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48966\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-25.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 25\" width=\"556\" height=\"139\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-25.png 556w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-25-300x75.png 300w\" sizes=\"(max-width: 556px) 100vw, 556px\" \/><\/p>\n<p>Question 26.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41707\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-58.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 58\" width=\"325\" height=\"138\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-58.png 325w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-58-300x127.png 300w\" sizes=\"(max-width: 325px) 100vw, 325px\" \/><\/p>\n<p>Answer:<br \/>\nBecause point T is the intersection of the angle bisectors, it is the incenter. But, because TV and TF are not necessarily perpendicular to a side of the trinagle, there is not sufficient evidence to conclude that TV and TZ are congruent. Point T is equidistance from the sides of the triangle.<\/p>\n<p>Question 27.<br \/>\n<strong>MODELING WITH MATHEMATICS<\/strong><br \/>\nYou and two friends plan to meet to walk your dogs together. You want the meeting place to be the same distance from each person\u2019s house. Explain how you can use the diagram to locate the meeting place.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41706\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-59.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 59\" width=\"315\" height=\"127\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-59.png 315w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-59-300x121.png 300w\" sizes=\"(max-width: 315px) 100vw, 315px\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48967\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-27.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 27\" width=\"570\" height=\"142\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-27.png 570w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-27-300x75.png 300w\" sizes=\"(max-width: 570px) 100vw, 570px\" \/><\/p>\n<p>Question 28.<br \/>\n<strong>MODELING WITH MATHEMATICS<\/strong><br \/>\nYou are placing a fountain in a triangular koi pond. YOU want the foutain to be the same distance from each edge of the Pond. Where should you place the fountain? Explain your reasoning. Use a sketch to support your answer.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41705\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-60.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 60\" width=\"322\" height=\"164\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-60.png 322w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-60-300x153.png 300w\" sizes=\"(max-width: 322px) 100vw, 322px\" \/><\/p>\n<p>Answer:<br \/>\nA, B, C are edges of the pond.<br \/>\nFoutain to be the same distance from each edge of the Pond<br \/>\nThen, you need to find the circumcenter of the triangle.<br \/>\nPlace the fountain at the circumcenter.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-105514\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-geometry-answers-chapter-6-relationships-within-triangles-6.2-6.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 6\" width=\"377\" height=\"272\" \/><\/p>\n<p><strong>CRITICAL THINKING<\/strong><br \/>\nIn Exercises 29-32, complete the statement with always, sometimes, or never. Explain your reasoning.<\/p>\n<p>Question 29.<br \/>\nThe circumenter of a scalene triangle is ______________ inside the triangle.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48968\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-29.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 29\" width=\"582\" height=\"142\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-29.png 582w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-29-300x73.png 300w\" sizes=\"(max-width: 582px) 100vw, 582px\" \/><\/p>\n<p>Question 30.<br \/>\nIf the perpendicular bisector of one side of a triangle intersects the opposite vertex. then the triangle is ______________ isosceles.<\/p>\n<p>Answer:<br \/>\nIf the perpendicular bisector of one side of a triangle intersects the opposite vertex. then the triangle is always isosceles.<\/p>\n<p>Explanation:<br \/>\nThe Vertex is equidistant to other vertices then.<\/p>\n<p>Question 31.<br \/>\nThe perpendicular bisectors of a triangle intersect at a point that is ______________ equidistant from the midpoints of the sides of the triangle.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48969\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-31.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 31\" width=\"567\" height=\"113\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-31.png 567w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-31-300x60.png 300w\" sizes=\"(max-width: 567px) 100vw, 567px\" \/><\/p>\n<p>Question 32.<br \/>\nThe angle bisectors of a triangle intersect at a point that is ______________ equidistant from the sides of the triangle.<\/p>\n<p>Answer:<br \/>\nThe angle bisectors of a triangle intersect at a point that is circumcenter equidistant from the sides of the triangle.<\/p>\n<p><strong>CRITICAL THINKING<\/strong><br \/>\nIn Exercises 33 and 34, find the coordinates of the circumcenter of the triangle with the given vertices.<\/p>\n<p>Question 33.<br \/>\nA(2, 5), B(6, 6). C( 12. 3)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48970\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-33.1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 33.1\" width=\"423\" height=\"464\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-33.1.png 423w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-33.1-273x300.png 273w\" sizes=\"(max-width: 423px) 100vw, 423px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48971\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-33.2.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 33.2\" width=\"490\" height=\"512\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-33.2.png 490w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-33.2-287x300.png 287w\" sizes=\"(max-width: 490px) 100vw, 490px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48972\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-33.3.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 33.3\" width=\"514\" height=\"288\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-33.3.png 514w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-33.3-300x168.png 300w\" sizes=\"(max-width: 514px) 100vw, 514px\" \/><\/p>\n<p>Question 34.<br \/>\nD(- 9, &#8211; 5), E(- 5, &#8211; 9), F(- 2, &#8211; 2)<\/p>\n<p>Answer:<br \/>\nThe circumcenter is (4.45, 4.45)<\/p>\n<p>Explanation:<br \/>\nRecall that the circumcentre of a triangle is equidistant from the vertices of a triangle. Let D(- 9, &#8211; 5), E(- 5, &#8211; 9), F(- 2, &#8211; 2) be the vertices of the given triangle and let P(x,y) be the circumcentre of this triangle. Then<br \/>\nPD = PE = PF<br \/>\nPD\u00b2 = PE\u00b2 = PF\u00b2<br \/>\nPD\u00b2 = PE\u00b2<br \/>\n(x + 9)\u00b2 + (y + 5)\u00b2 = (x + 5)\u00b2 + (y + 9)\u00b2<br \/>\nx\u00b2 + 18x + 81 + y\u00b2 + 10y + 25 = x\u00b2 + 10x + 25 + y\u00b2 + 18y + 81<br \/>\n18x &#8211; 10x = 18y &#8211; 10y<br \/>\n8x = 8y<br \/>\nx = y<br \/>\nPE\u00b2 = PF\u00b2<br \/>\n(x + 5)\u00b2 + (y + 9)\u00b2 = (x + 2)\u00b2 + (y + 2)\u00b2<br \/>\nx\u00b2 + 10x + 25 + y\u00b2 + 18y + 81 = x\u00b2 + 4x + 4 + y\u00b2 + 2y + 4<br \/>\n10x &#8211; 4x + 106 = 8 + 2y &#8211; 18y<br \/>\n6x + 106 = 8 &#8211; 16y<br \/>\n6x + 98 = -16x<br \/>\n22x = -98<br \/>\nx = 4.45<br \/>\ny = 4.45<br \/>\nThe circumcenter is (4.45, 4.45)<\/p>\n<p><strong>MATHEMATICAL CONNECTIONS<\/strong><br \/>\nIn Exercises 35 and 36. find the a1ue of x that makes N the incenter of the triangle.<\/p>\n<p>Question 35.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41704\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-61.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 61\" width=\"187\" height=\"132\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48973\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-35.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 35\" width=\"255\" height=\"227\" \/><\/p>\n<p>Question 36.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41703\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-62.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 62\" width=\"186\" height=\"136\" \/><\/p>\n<p>Answer:<br \/>\nN the incenter is \\(\\frac { 12 }{ 2 } \\).<\/p>\n<p>Explanation:<br \/>\n25\u00b2 = 24\u00b2 + (14x)\u00b2<br \/>\n625 = 576 + 196x\u00b2<br \/>\n49 = 196x\u00b2<br \/>\nx = \\(\\frac { 7 }{ 14 } \\) = \\(\\frac { 1 }{ 2 } \\)<br \/>\nThe value of x will make N the incenter is \\(\\frac { 12 }{ 2 } \\).<\/p>\n<p>Question 37.<br \/>\n<strong>PROOF<\/strong><br \/>\nWhere is the circumcenter located in any right triangle? Write a coordinate proof of this result.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48974\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-37.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 37\" width=\"580\" height=\"568\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-37.png 580w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-37-300x294.png 300w\" sizes=\"(max-width: 580px) 100vw, 580px\" \/><\/p>\n<p>Question 38.<br \/>\n<strong>PROVING A THEOREM<\/strong><br \/>\nWrite a proof of the Incenter Theorem (Theorem 6.6).<br \/>\nGiven \u2206ABC, \\(\\overline{A D}\\) bisects\u2220CAB,<br \/>\n\\(\\overline{B D}\\) bisects \u2220CBA, \\(\\overline{D E}\\) \u22a5 \\(\\overline{A B}\\), \\(\\overline{D F}\\) \u22a5 \\(\\overline{B C}\\), and \\(\\overline{D G}\\) \u22a5 \\(\\overline{C A}\\)<br \/>\nProve The angle bisectors intersect at D, which is equidistant from \\(\\overline{A B}\\), \\(\\overline{B C}\\), and \\(\\overline{C A}\\)<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41702\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-63.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 63\" width=\"200\" height=\"132\" \/><\/p>\n<p>Answer:<\/p>\n<p>Question 39.<br \/>\n<strong>WRITING<\/strong><br \/>\nExplain the difference between the circumcenter and the incenter of a triangle.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48975\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-39.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 39\" width=\"562\" height=\"147\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-39.png 562w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-39-300x78.png 300w\" sizes=\"(max-width: 562px) 100vw, 562px\" \/><\/p>\n<p>Question 40.<br \/>\n<strong>REASONING<\/strong><br \/>\nIs the incenter of a triangle ever located outside the triangle? Explain your reasoning.<\/p>\n<p>Answer:<br \/>\nNo<\/p>\n<p>Explanation:<br \/>\nBecause the incenter is the center of an inscribed circle, it must always be inside the triangle.<\/p>\n<p>Question 41.<br \/>\n<strong>MODELING WITH MATHEMATICS<\/strong><br \/>\nYou are installing a circular pool in the triangular courtyard shown. You want to have the largest pool possible on the site without extending into the walkway.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41701\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-64.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 64\" width=\"306\" height=\"189\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-64.png 306w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-64-300x185.png 300w\" sizes=\"(max-width: 306px) 100vw, 306px\" \/><br \/>\na. Copy the triangle and show how to install the pool so that it just touches each edge. Then explain how you can he sure that you could not fit a larger pool on the site.<br \/>\nb. You want to have the largest pool possible while leaving at least I foot of space around the pool. Would the center of the pool be in the same position as in part (a)? Justify your answer.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48976\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-41.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 41\" width=\"607\" height=\"558\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-41.png 607w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-41-300x276.png 300w\" sizes=\"(max-width: 607px) 100vw, 607px\" \/><\/p>\n<p>Question 42.<br \/>\n<strong>MODELING WITH MATHEMATICS<\/strong><br \/>\nArchaeologists find three stones. They believe that the stones were once pail of a circle of stones with a community fire pit at its center. They mark the locations of stones A, B, and C on a graph. where distances are measured in feet.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41700\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-65.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 65\" width=\"268\" height=\"217\" \/><br \/>\na. Explain how archaeologists can use a sketch to estimate the center of the circle of stones.<\/p>\n<p>b. Copy the diagram and find the approximate coordinates of the point at which the archaeologists should look for the fire pit.<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-106566\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-answers-geometry-chapter-6-relationships-within-triangles-6.2-3.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 6.2 3\" width=\"362\" height=\"340\" \/><br \/>\nTo get the centre of the circle P, we need to find the circumcenter of the triangle ABC.<br \/>\nRecall that the circumcentre of a triangle is equidistant from the vertices of a triangle. Let A(2, 10), B(13, 6), C(6, 1) be the vertices of the given triangle and let P(x,y) be the circumcentre of this triangle. Then<br \/>\nPA = PB = PC<br \/>\nPA\u00b2 = PB\u00b2 = PC\u00b2<br \/>\nPA\u00b2 = PB\u00b2<br \/>\n(x &#8211; 2)\u00b2 + (y &#8211; 10)\u00b2 = (x &#8211; 13)\u00b2 + (y &#8211; 6)\u00b2<br \/>\nx\u00b2 &#8211; 4x + 4 + y\u00b2 &#8211; 20y + 100 = x\u00b2 &#8211; 26x + 169 + y\u00b2 &#8211; 12y + 36<br \/>\n-4x + 26x &#8211; 20y + 12y = 205 &#8211; 104<br \/>\n22x &#8211; 8y = 101 &#8212;- (i)<br \/>\nPB\u00b2 = PC\u00b2<br \/>\n(x &#8211; 13)\u00b2 + (y &#8211; 6)\u00b2 = (x &#8211; 8)\u00b2 + (y &#8211; 1)\u00b2<br \/>\nx\u00b2 &#8211; 26x + 169 + y\u00b2 &#8211; 12y + 36 = x\u00b2 &#8211; 16x + 64 + y\u00b2 &#8211; 2y + 1<br \/>\n-26x + 16x &#8211; 12y + 2y = 65 &#8211; 205<br \/>\n-10x &#8211; 10y = -140<br \/>\nx + y = 14<br \/>\nx = 14 &#8211; y &#8212; (ii)<br \/>\nSubstitute (ii) in (i)<br \/>\n22(14 &#8211; y) &#8211; 8y = 101<br \/>\n308 &#8211; 22y &#8211; 8y = 101<br \/>\n308 &#8211; 101 = 30y<br \/>\n207 = 30y<br \/>\ny = 6.9<br \/>\nSubstitute y = 6.9 in (ii)<br \/>\nx = 14 &#8211; 6.9<br \/>\nx = 7.1<br \/>\nThe circumcenter is (7.1, 6.9)<\/p>\n<p>Question 43.<br \/>\n<strong>REASONING<\/strong><br \/>\nPoint P is inside \u2206ABC and is equidistant from points A and B. On which of the following segments must P be located?<br \/>\n(A) \\(\\overline{A B}\\)<br \/>\n(B) the perpendicular bisector of \\(\\overline{A B}\\)<br \/>\n(C) \\(\\overline{A C}\\)<br \/>\n(D) the perpendicular bisector of \\(\\overline{A C}\\)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48977\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-43.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 43\" width=\"390\" height=\"33\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-43.png 390w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-43-300x25.png 300w\" sizes=\"(max-width: 390px) 100vw, 390px\" \/><\/p>\n<p>Question 44.<br \/>\n<strong>CRITICAL THINKING<\/strong><br \/>\nA high school is being built for the four towns shown on the map. Each town agrees that the school should be an equal distance from each of the tourist towns. Is there a single point where they could agree to build the school? If so, find it. If not, explain why not. Justify your answer with a diagram.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41699\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-66.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 66\" width=\"232\" height=\"126\" \/><br \/>\nAnswer:<\/p>\n<p>Question 45.<br \/>\n<strong>MAKING AN ARGUMENT<\/strong><br \/>\nYour friend says that the circumcenter of an equilateral triangle is also the incenter of the triangle. Is your friend correct? Explain in your reasoning.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48978\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-45.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 45\" width=\"581\" height=\"84\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-45.png 581w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-45-300x43.png 300w\" sizes=\"(max-width: 581px) 100vw, 581px\" \/><\/p>\n<p>Question 46.<br \/>\n<strong>HOW DO YOU SEE IT?<\/strong><br \/>\nThe arms of the windmill are the angle bisectors of the red triangle. What point of concurrency is the point that Connects the three arms?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41698\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-67.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 67\" width=\"212\" height=\"186\" \/><br \/>\nAnswer:<\/p>\n<p>Question 47.<br \/>\n<strong>ABSTRACT REASONING<\/strong><br \/>\nYou are asked to draw a triangle and all its perpendicular bisectors and angle bisectors.<br \/>\na. For which type of triangle would you need the fewest segments? What is the minimum number of segments you would need? Explain.<br \/>\nb. For which type of triangle would you need the most segments? What is the maximum number of segments you would need? Explain.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48979\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-47.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 47\" width=\"573\" height=\"233\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-47.png 573w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-47-300x122.png 300w\" sizes=\"(max-width: 573px) 100vw, 573px\" \/><\/p>\n<p>Question 48.<br \/>\n<strong>THOUGHT PROVOKING<\/strong><br \/>\nThe diagram shows an official hockey rink used by the National Hockey League. Create a triangle using hockey players as vertices in which the center circle is inscribed in the triangle. The center dot should he the incenter of your triangle. Sketch a drawing of the locations of your hockey players. Then label the actual lengths of the sides and the angle measures in your triangle.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41697\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-68.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 68\" width=\"310\" height=\"148\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-68.png 310w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-68-300x143.png 300w\" sizes=\"(max-width: 310px) 100vw, 310px\" \/><br \/>\nAnswer:<\/p>\n<p><strong>COMPARING METHODS<\/strong><br \/>\nIn Exercises 49 and 50. state whether you would use perpendicular bisectors or angle bisectors. Then solve the problem.<\/p>\n<p>Question 49.<br \/>\nYou need to cut the largest circle possible from an isosceles triangle made of paper whose sides are 8 inches, 12 inches, and 12 inches. Find the radius of the circle.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48980\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-49.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 49\" width=\"593\" height=\"407\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-49.png 593w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-49-300x206.png 300w\" sizes=\"(max-width: 593px) 100vw, 593px\" \/><\/p>\n<p>Question 50.<br \/>\nOn a map of a camp. You need to create a circular walking path that connects the pool at (10, 20), the nature center at (16, 2). and the tennis court at (2, 4). Find the coordinates of the center of the circle and the radius of the circle.<\/p>\n<p>Answer:<br \/>\nThe center of the circular path is located at (10, 10) and the radius of the circular path is 10 units.<\/p>\n<p>Explanation:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-105530\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-geometry-answers-chapter-6-relationships-within-triangles-6.2-7.jpg\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 7\" width=\"634\" height=\"589\" \/><\/p>\n<p>Let the centre of the circle be at O (x, y)<br \/>\nSlope of AB = \\(\\frac { 20 &#8211; 4 }{ 10 &#8211; 2 } \\) = 2<br \/>\nThe slope of XO must be \\(\\frac { -1}{ 2 } \\) the negative reciprocal of the slope of AB as the 2 lines are perpendicular<br \/>\nSlope of XO = \\(\\frac { y &#8211; 12 }{ x &#8211; 6 } \\) = \\(\\frac { -1 }{ 2 } \\)<br \/>\ny &#8211; 12 = -0.5x + 3<br \/>\n0.5x + y = 12 + 3 = 15<br \/>\nx + 2y = 30<br \/>\nThe slope of BC = \\(\\frac { 2 &#8211; 20 }{ 16 &#8211; 10 } \\) = -3<br \/>\nThe slope of XO must be \\(\\frac { 1 }{ 3 } \\) = \\(\\frac { 11 &#8211; y }{ 13 &#8211; x } \\)<br \/>\n33 &#8211; 3y = 13 &#8211; x<br \/>\nx &#8211; 3y = -33 + 13 = -20<br \/>\nSubtrcat two equations<br \/>\nx + 2y &#8211; x + 3y = 30 + 20<br \/>\ny = 10<br \/>\nx &#8211; 30 = -20<br \/>\nx = 10<br \/>\nr = \u221a(10 &#8211; 2)\u00b2 + (10 &#8211; 4)\u00b2<br \/>\nr = 10<\/p>\n<p>Question 51.<br \/>\n<strong>CRITICAL THINKING<\/strong><br \/>\nPoint D is the incenter of \u2206ABC. Write an expression for the length x in terms of the three side lengths AB, AC, and BC.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-41696\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-69.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 69\" width=\"262\" height=\"105\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48981\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-51.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 51\" width=\"567\" height=\"576\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-51.png 567w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-51-295x300.png 295w\" sizes=\"(max-width: 567px) 100vw, 567px\" \/><\/p>\n<p>Maintaining Mathematical Proficiency<\/p>\n<p>The endpoints of \\(\\overline{A B}\\) are given. Find the coordinates of the midpoint M. Then find AB.<br \/>\nQuestion 52.<br \/>\nA(- 3, 5), B(3, 5)<\/p>\n<p>Answer:<br \/>\nAB = 6<\/p>\n<p>Explanation:<br \/>\nMidpoint of AB = (\\(\\frac { -3 + 3 }{ 2 } \\), \\(\\frac { 5 + 5 }{ 2 } \\)) = (0, 5)<br \/>\nAB = \u221a(3 + 3)\u00b2 + (5 &#8211; 5)\u00b2 = 6<\/p>\n<p>Question 53.<br \/>\nA(2, &#8211; 1), B(10, 7)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48982\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-53.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 53\" width=\"527\" height=\"130\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-53.png 527w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-53-300x74.png 300w\" sizes=\"(max-width: 527px) 100vw, 527px\" \/><\/p>\n<p>Question 54.<br \/>\nA(- 5, 1), B(4, &#8211; 5)<\/p>\n<p>Answer:<br \/>\nAB = 10.81<\/p>\n<p>Explanation:<br \/>\nMidpoint of AB = (\\(\\frac { -5 + 4 }{ 2 } \\), \\(\\frac { 1 &#8211; 5 }{ 2 } \\)) = (\\(\\frac { -1 }{ 2 } \\), -2)<br \/>\nAB = \u221a(4 + 5)\u00b2 + (-5 &#8211; 1)\u00b2 = \u221a81 + 36 = \u00a010.81<\/p>\n<p>Question 55.<br \/>\nA(- 7, 5), B(5, 9)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48983\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-55.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 55\" width=\"556\" height=\"133\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-55.png 556w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-55-300x72.png 300w\" sizes=\"(max-width: 556px) 100vw, 556px\" \/><\/p>\n<p>Write an equation of the line passing through point P that is perpendicular to the given line.<br \/>\nGraph the equations of the lines to check that they are perpendicular.<br \/>\nQuestion 56.<br \/>\nP(2, 8), y = 2x + 1<\/p>\n<p>Answer:<br \/>\ny = \\(\\frac { -1 }{ 2 } \\)x + 9<\/p>\n<p>Explanation:<br \/>\nThe slope of the given line m = 2<br \/>\nThe slope of the perpendicular line M = \\(\\frac { -1 }{ 2 } \\)<br \/>\nThe perpendicular line passes through the given point P(2, 8) is<br \/>\n8 = \\(\\frac { -1 }{ 2 } \\)(2) + b<br \/>\nb = 9<br \/>\nSo, y = \\(\\frac { -1 }{ 2 } \\)x + 9<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-106577\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-answers-geometry-chapter-6-relationships-within-triangles-6.2-4.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 6.2 4\" width=\"1000\" height=\"896\" \/><\/p>\n<p>Question 57.<br \/>\nP(6, -3), y = &#8211; 5<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48984\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-57.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 57\" width=\"538\" height=\"361\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-57.png 538w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-57-300x201.png 300w\" sizes=\"(max-width: 538px) 100vw, 538px\" \/><\/p>\n<p>Question 58.<br \/>\nP(- 8, &#8211; 6), 2x + 3y = 18<\/p>\n<p>Answer:<br \/>\nThe perpendicular equation is y = \\(\\frac { 3 }{ 2 } \\)x + 6<\/p>\n<p>Explanation:<br \/>\nThe given line is 2x + 3y = 18<br \/>\ny = \\(\\frac { -2 }{ 3 } \\)x + 6<br \/>\nThe slope of the given line m = \\(\\frac { -2 }{ 3 } \\)<br \/>\nThe slope of the perpendicular line is M = \\(\\frac { 3 }{ 2 } \\)<br \/>\nThe perpendicular line passing through the point P(-8, -6) is<br \/>\n-6 = \\(\\frac { 3 }{ 2 } \\)(-8) + b<br \/>\nb = 6<br \/>\nThe perpendicular equation is y = \\(\\frac { 3 }{ 2 } \\)x + 6<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-106591\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-answers-geometry-chapter-6-relationships-within-triangles-6.2-5.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 6.2 5\" width=\"1000\" height=\"896\" \/><\/p>\n<p>Question 59.<br \/>\nP(- 4, 1), y + 3 = &#8211; 4(x + 3)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48985\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-59.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.2 Question 59\" width=\"571\" height=\"459\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-59.png 571w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.2-Question-59-300x241.png 300w\" sizes=\"(max-width: 571px) 100vw, 571px\" \/><\/p>\n<h3><a id=\"6.3_Medians_and_Altitudes_of_Triangles\"><\/a>6.3 Medians and Altitudes of Triangles<\/h3>\n<p><strong>Exploration 1<\/strong><\/p>\n<p>Finding Properties of the Medians of a Triangle<\/p>\n<p>Work with a partner. Use dynamic geometry software. Draw any \u2206ABC.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-42249\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-70.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 70\" width=\"407\" height=\"240\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-70.png 407w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-70-300x177.png 300w\" sizes=\"(max-width: 407px) 100vw, 407px\" \/><\/p>\n<p>a. Plot the midpoint of \\(\\overline{B C}\\) and label it D, Draw \\(\\overline{A D}\\), which is a median of ABC. Construct the medians to the other two sides of \u2206ABC.<br \/>\nAnswer:<\/p>\n<p>b. What do you notice about the medians? Drag the vertices to change \u2206ABC. Use your observations to write a conjecture about the medians of a triangle.<br \/>\nAnswer:<\/p>\n<p>c. In the figure above, point G divides each median into a shorter segment and a longer segment. Find the ratio of the length of each longer segment to the length of the whole median. Is this ratio always the same? Justify your answer.<br \/>\nAnswer:<\/p>\n<p><strong>Exploration 2<\/strong><\/p>\n<p>Finding Properties of the Altitudes of a Triangle<\/p>\n<p>Work with a partner. Use dynamic geometry software. Draw any \u2206ABC.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-42248\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-71.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 71\" width=\"302\" height=\"241\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-71.png 302w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-71-300x239.png 300w\" sizes=\"(max-width: 302px) 100vw, 302px\" \/><\/p>\n<p>a. Construct the perpendicular segment from vertex A to \\(\\overline{B C}\\). Label the endpoint D. \\(\\overline{A D}\\) is an altitude of \u2206ABC.<br \/>\nAnswer:<\/p>\n<p>b. Construct the altitudes to the other two sides of \u2206ABC. What do you notice?<br \/>\nAnswer:<\/p>\n<p>c. Write a conjecture about the altitudes of a triangle. Test your conjecture by dragging the vertices to change \u2206ABC.<br \/>\n<strong>LOOKING FOR STRUCTURE<\/strong><br \/>\nTo be proficient in math, you need to look closely to discern a pattern or structure.<br \/>\nAnswer:<\/p>\n<p>Communicate Your Answer<\/p>\n<p>Question 3.<br \/>\nWhat conjectures can you make about the medians and altitudes of a triangle?<br \/>\nAnswer:<\/p>\n<p>Question 4.<br \/>\nThe length of median \\(\\overline{R U}\\) in \u2206RST is 3 inches. The point 0f concurrency of the three medians of \u2206RST divides \\(\\overline{R U}\\) into two segments. What are the lengths of these two segments?<br \/>\nAnswer:<\/p>\n<h3><a id=\"Lesson_6.3_Medians_and_Altitudes_of_Triangles\"><\/a>Lesson 6.3 Medians and Altitudes of Triangles<\/h3>\n<p><strong>Monitoring Progress<\/strong><\/p>\n<p>There are three paths through a triangular park. Each path goes from the midpoint of one edge to the opposite corner. The paths meet at point P.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-42247\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-72.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 72\" width=\"234\" height=\"173\" \/><\/p>\n<p>Question 1.<br \/>\nFind PS and PC when SC = 2100 feet.<\/p>\n<p>Answer:<br \/>\nPS = 700, PC = 2100<\/p>\n<p>Explanation:<br \/>\nPC = \\(\\frac { 2 }{ 3 } \\)SC<br \/>\nSC = PS + PC<br \/>\nSC = PS + \\(\\frac { 2 }{ 3 } \\)SC<br \/>\nPS = \\(\\frac { 1 }{ 3 } \\)SC<br \/>\nSo, PS = \\(\\frac { 1 }{ 3 } \\)(2100) = 700<br \/>\nPC = \\(\\frac { 2 }{ 3 } \\)(2100) = 1400<\/p>\n<p>Question 2.<br \/>\nFind TC and BC when BT = 1000 feet.<\/p>\n<p>Answer:<br \/>\nBC = 2000 ft, TC = 1000 ft<\/p>\n<p>Explanation:<br \/>\nBT = TC<br \/>\nSo, TC = 1000 ft<br \/>\nBC = BT + TC<br \/>\nBC = 1000 + 1000 = 2000 ft<\/p>\n<p>Question 3.<br \/>\nFind PA and TA when PT = 800 feet.<\/p>\n<p>Answer:<br \/>\nPA = 2400 ft, TA = 1600 ft<\/p>\n<p>Explanation:<br \/>\nPT = \\(\\frac { 1 }{ 3 } \\)PA<br \/>\nPA = 3PT<br \/>\n= 3 \u2022 800 = 2400<br \/>\nTA = \\(\\frac { 2 }{ 3 } \\)PA<br \/>\nTA = \\(\\frac { 2 }{ 3 } \\)(2400)<br \/>\n= 1600<\/p>\n<p>Find the coordinates of the centroid of the triangle with the given vertices.<\/p>\n<p>Question 4.<br \/>\nF(2, 5), G(4, 9), H(6, 1)<\/p>\n<p>Answer:<br \/>\nThe centroid is (4, 5).<\/p>\n<p>Explanation:<br \/>\nThe centroid of the triangle FGH is O = (\\(\\frac { 2 + 4 + 6 }{ 3 } \\), \\(\\frac { 5 + 9 + 1 }{ 3 } \\))<br \/>\n= (\\(\\frac { 12 }{ 3 } \\), \\(\\frac { 15 }{ 3 } \\))<br \/>\n= (4, 5)<\/p>\n<p>Question 5.<br \/>\nX(- 3, 3), Y(1, 5), Z(- 1, &#8211; 2)<\/p>\n<p>Answer:<br \/>\nThe centriod of given triangle is (-1, 2)<\/p>\n<p>Explanation:<br \/>\nThe centroid of the triangle XYZ is O = (\\(\\frac { -3 + 1 &#8211; 1 }{ 3 } \\), \\(\\frac { 3 + 5 &#8211; 2 }{ 3 } \\))<br \/>\n= (\\(\\frac { -3 }{ 3 } \\), \\(\\frac { 6 }{ 3 } \\))<br \/>\n= (-1, 2)<\/p>\n<p>Tell whether the orthocenter of the triangle with the given vertices is inside, on, or outside the triangle. Then find the coordinates of the orthocenter.<\/p>\n<p>Question 6.<br \/>\nA(0, 3), B(0, &#8211; 2), C(6, -3)<\/p>\n<p>Answer:<br \/>\nThe orthocenter lies outside the triangle.<\/p>\n<p>Explanation:<br \/>\nThe slope of the line BC = \\(\\frac { -3 + 2 }{ 6 &#8211; 0 } \\) = \\(\\frac { -1 }{ 6 } \\)<br \/>\nThe slope of the perpendicular line = 6<br \/>\nThe perpendicular line is (y &#8211; 3) = 6(x &#8211; 0)<br \/>\ny &#8211; 3 = 6x<br \/>\ny = 6x + 3<br \/>\nThe slope of AC = \\(\\frac { -3 &#8211; 3 }{ 6 &#8211; 0 } \\) = \\(\\frac { -6 }{ 6 } \\) = -1<br \/>\nThe slope of the perpendicular line = 1<br \/>\nThe equation of perpendicular line (y + 2) = 1(x &#8211; 0)<br \/>\ny + 2 = x<br \/>\nSubstitute y = 6x + 3 in above equation<br \/>\n6x + 3 + 2 = x<br \/>\n5x = -5<br \/>\nx = -1<br \/>\nPut x = -1 in y + 2 = x<br \/>\ny + 2 = -1<br \/>\ny = -3<br \/>\nThe orthocenter is (-1, -3)<br \/>\nThe orthocenter lies outside the triangle.<\/p>\n<p>Question 7.<br \/>\nJ(- 3, &#8211; 4), K(- 3, 4), L(5, 4)<\/p>\n<p>Answer:<br \/>\nThe orthocenter lies on the vertex of the triangle.<\/p>\n<p>Explanation:<br \/>\nThe slope of JL = \\(\\frac { 4 + 4 }{ 5 + 3 } \\) = 2<br \/>\nThe slope of perpendicular line is \\(\\frac { -1 }{ 2 } \\)<br \/>\nThe equation of perpendicular line is (y &#8211; 4) = \\(\\frac { -1 }{ 2 } \\)(x + 3)<br \/>\n2y &#8211; 8 = -x &#8211; 3<br \/>\nx + 2y &#8211; 5 = 0<br \/>\nThe slope of JK = \\(\\frac { 4 + 4 }{ -3 + 3 } \\) = 0<br \/>\nThe slope of the perpendicular line is 0<br \/>\nthe equation of perpendicular line is (y &#8211; 4) = 0<br \/>\ny = 4<br \/>\nSubstitute y = 4 in x + 2y &#8211; 5 = 0<br \/>\nx + 2(4) &#8211; 5 = 0<br \/>\nx + 3 = 0<br \/>\nx = -3<br \/>\nSo, the ortho center is (-3, 4)<br \/>\nIt lies on the vertex of the triangle.<\/p>\n<p>Question 8.<br \/>\n<strong>WHAT IF?<\/strong><br \/>\nIn Example 4, you want to show that median \\(\\overline{B D}\\) is also an angle bisector. How would your proof be different?<br \/>\nAnswer:<\/p>\n<h3><a id=\"Exercise_6.3_Medians_and_Altitudes_of_Triangles\"><\/a>Exercise 6.3 Medians and Altitudes of Triangles<\/h3>\n<p>Vocabulary and Core Concept Check<\/p>\n<p>Question 1.<br \/>\n<strong>VOCABULARY<\/strong><br \/>\nName the four types of points of concurrency. Which lines intersect to form each of the points?<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48986\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 1\" width=\"585\" height=\"194\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-1.png 585w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-1-300x99.png 300w\" sizes=\"(max-width: 585px) 100vw, 585px\" \/><\/p>\n<p>Question 2.<br \/>\n<strong>COMPLETE THE SENTENCE<\/strong><br \/>\nThe length of a segment from a vertex to the centroid is ______________ the length of the median from that vertex.<\/p>\n<p>Answer:<br \/>\nThe length of a segment from a vertex to the centroid is one-third of the length of the median from that vertex.<\/p>\n<p>Monitoring progress and Modeling with Mathematics<\/p>\n<p>In Exercises 3-6, point P is the centroid of \u2206LMN. Find PN and QP.<\/p>\n<p>Question 3.<br \/>\nQN = 9<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42246\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-73.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 73\" width=\"147\" height=\"109\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48987\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-3.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 3\" width=\"160\" height=\"300\" \/><\/p>\n<p>Question 4.<br \/>\nQN = 21<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42245\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-74.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 74\" width=\"114\" height=\"113\" \/><\/p>\n<p>Answer:<br \/>\nPN = 14 units, QP = 7 units<\/p>\n<p>Explanation:<br \/>\nPN = \\(\\frac { 2 }{ 3 } \\)QN<br \/>\nPN = \\(\\frac { 2 }{ 3 } \\)(21)<br \/>\nPN = 14<br \/>\nQP = \\(\\frac { 1 }{ 3 } \\)QN<br \/>\n= \\(\\frac { 1 }{ 3 } \\)(21)<br \/>\n= 7<\/p>\n<p>Question 5.<br \/>\nQN = 30<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42244\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-75.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 75\" width=\"146\" height=\"107\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48988\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-5.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 5\" width=\"182\" height=\"300\" \/><\/p>\n<p>Question 6.<br \/>\nQN = 42<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42243\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-76.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 76\" width=\"163\" height=\"122\" \/><\/p>\n<p>Answer:<br \/>\nPN = 28 units, QP = 14 units<\/p>\n<p>Explanation:<br \/>\nPN = \\(\\frac { 2 }{ 3 } \\)QN<br \/>\nPN = \\(\\frac { 2 }{ 3 } \\)(42)<br \/>\nPN = 28<br \/>\nQP = \\(\\frac { 1 }{ 3 } \\)QN<br \/>\n= \\(\\frac { 1 }{ 3 } \\)(42)<br \/>\n= 14<\/p>\n<p>In Exercises 7-10. point D is the centroid of \u2206 ABC. Find CD and CE.<\/p>\n<p>Question 7.<br \/>\nDE = 5<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42242\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-77.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 77\" width=\"96\" height=\"124\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48989\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-7.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 7\" width=\"206\" height=\"334\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-7.png 206w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-7-185x300.png 185w\" sizes=\"(max-width: 206px) 100vw, 206px\" \/><\/p>\n<p>Question 8.<br \/>\nDE = 11<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42241\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-78.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 78\" width=\"137\" height=\"134\" \/><\/p>\n<p>Answer:<br \/>\nCD = 22 units, CE = 33 units<\/p>\n<p>Explanation:<br \/>\nDE = \\(\\frac { 1 }{ 3 } \\)CE<br \/>\n11 = \\(\\frac { 1 }{ 3 } \\) CE<br \/>\nCE = 33<br \/>\nCD = \\(\\frac { 2 }{ 3 } \\) CE<br \/>\nCD = \\(\\frac { 2 }{ 3 } \\)(33)<br \/>\nCD = 22<\/p>\n<p>Question 9.<br \/>\nDE = 9<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42240\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-79.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 79\" width=\"138\" height=\"137\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48990\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-9.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 9\" width=\"200\" height=\"337\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-9.png 200w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-9-178x300.png 178w\" sizes=\"(max-width: 200px) 100vw, 200px\" \/><\/p>\n<p>Question 10.<br \/>\nDE = 15<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42239\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-80.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 80\" width=\"160\" height=\"126\" \/><\/p>\n<p>Answer:<br \/>\nCE = 45 units, CD = 30<\/p>\n<p>Explanation:<br \/>\nDE = \\(\\frac { 1 }{ 3 } \\)CE<br \/>\n15 = \\(\\frac { 1 }{ 3 } \\) CE<br \/>\nCE = 45<br \/>\nCD = \\(\\frac { 2 }{ 3 } \\) CE<br \/>\nCD = \\(\\frac { 2 }{ 3 } \\)(45)<br \/>\nCD = 30<\/p>\n<p>In Exercises 11-14. point G is the centroid of \u2206ABC. BG = 6, AF = 12, and AE = 15. Find the length of the segment.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-42238\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-81.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 81\" width=\"291\" height=\"101\" \/><\/p>\n<p>Question 11.<br \/>\n\\(\\overline{F C}\\)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48991\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-11.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 11\" width=\"572\" height=\"67\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-11.png 572w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-11-300x35.png 300w\" sizes=\"(max-width: 572px) 100vw, 572px\" \/><\/p>\n<p>Question 12.<br \/>\n\\(\\overline{B F}\\)<\/p>\n<p>Answer:<br \/>\n\\(\\overline{B F}\\) = 15<\/p>\n<p>Explanation:<br \/>\nAll the medians have same length.<br \/>\nso, AE = 15 = BF<\/p>\n<p>Question 13.<br \/>\n\\(\\overline{A G}\\)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48992\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-13.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 13\" width=\"559\" height=\"180\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-13.png 559w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-13-300x97.png 300w\" sizes=\"(max-width: 559px) 100vw, 559px\" \/><\/p>\n<p>Question 14.<br \/>\n\\(\\overline{G E}\\)<\/p>\n<p>Answer:<br \/>\n\\(\\overline{G E}\\) = 5 units<\/p>\n<p>Explanation:<br \/>\nAE = AG + GE<br \/>\n15 = 10 + GE<br \/>\nGE = 5<\/p>\n<p>In Exercises 15-18. find the coordinates of the centroid of the triangle with the given vertices.<\/p>\n<p>Question 15.<br \/>\nA(2, 3), B(8, 1), C(5, 7)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48993\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-15.1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 15.1\" width=\"559\" height=\"448\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-15.1.png 559w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-15.1-300x240.png 300w\" sizes=\"(max-width: 559px) 100vw, 559px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48994\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-15.2.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 15.2\" width=\"564\" height=\"491\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-15.2.png 564w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-15.2-300x261.png 300w\" sizes=\"(max-width: 564px) 100vw, 564px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48995\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-15.3.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 15.3\" width=\"348\" height=\"272\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-15.3.png 348w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-15.3-300x234.png 300w\" sizes=\"(max-width: 348px) 100vw, 348px\" \/><\/p>\n<p>Question 16.<br \/>\nF(1, 5), G( &#8211; 2, 7), H(- 6, 3)<\/p>\n<p>Answer:<br \/>\nThe centroid is (\\(\\frac { -7 }{ 3 } \\), 5)<\/p>\n<p>Explanation:<br \/>\nThe centroid of the trinagle = (\\(\\frac { 1 &#8211; 2 &#8211; 6 }{ 3 } \\), \\(\\frac { 5 + 7 + 3 }{ 3 } \\))<br \/>\n= (\\(\\frac { -7 }{ 3 } \\), 5)<\/p>\n<p>Question 17.<br \/>\nS(5, 5), T(11, &#8211; 3), U(- 1, I)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48996\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-17.1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 17.1\" width=\"567\" height=\"498\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-17.1.png 567w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-17.1-300x263.png 300w\" sizes=\"(max-width: 567px) 100vw, 567px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48997\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-17.2.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 17.2\" width=\"560\" height=\"137\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-17.2.png 560w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-17.2-300x73.png 300w\" sizes=\"(max-width: 560px) 100vw, 560px\" \/><\/p>\n<p>Question 18.<br \/>\nX(1, 4), Y(7, 2), Z(2, 3)<\/p>\n<p>Answer:<br \/>\nThe centroid of the triangle is (\\(\\frac { 10 }{ 2 } \\), 3)<\/p>\n<p>Explanation:<br \/>\nThe centroid of the trinagle = (\\(\\frac { 1 + 7 + 2 }{ 3 } \\), \\(\\frac { 4 + 2 + 3 }{ 3 } \\))<br \/>\n= (\\(\\frac { 10 }{ 2 } \\), 3)<\/p>\n<p>In Exercises 19-22. tell whether the orthocenter is inside, on, or outside the triangle. Then find the coordinates of the orthocenter.<\/p>\n<p>Question 19.<br \/>\nL(0, 5), M(3, 1), N(8, 1)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48998\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-19.1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 19.1\" width=\"558\" height=\"489\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-19.1.png 558w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-19.1-300x263.png 300w\" sizes=\"(max-width: 558px) 100vw, 558px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-48999\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-19.2.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 19.2\" width=\"594\" height=\"472\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-19.2.png 594w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-19.2-300x238.png 300w\" sizes=\"(max-width: 594px) 100vw, 594px\" \/><\/p>\n<p>Question 20.<br \/>\nX(- 3, 2), Y(5, 2), Z(- 3, 6)<\/p>\n<p>Answer:<br \/>\nThe orthocenter lies on the vertex of the triangle.<\/p>\n<p>Explanation:<br \/>\nThe slope of YZ = \\(\\frac { 6 &#8211; 2 }{ -3 &#8211; 5 } \\) = \\(\\frac { -1 }{ 2 } \\)<br \/>\nThe slope of the perpendicular line is 2<br \/>\nThe equation of perpendicular line is (y &#8211; 2) = 2(x + 3)<br \/>\ny &#8211; 2 = 2x + 6<br \/>\n2x &#8211; y + 8 = 0<br \/>\nThe slope of XZ = \\(\\frac { 6 &#8211; 2 }{ -3 + 3 } \\) = 0<br \/>\nThe equation of perpendicular line is (y &#8211; 2) = 0<br \/>\ny = 2<br \/>\nSubstitute y = 2 in 2x &#8211; y + 8 = 0<br \/>\n2x &#8211; 2 + 8 = 0<br \/>\n2x + 6 = 0<br \/>\nx = -3<br \/>\nthe orthocenter is (-3, 2)<br \/>\nThe orthocenter lies on the vertex of the triangle.<\/p>\n<p>Question 21.<br \/>\nA(- 4, 0), B(1, 0), C(- 1, 3)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49000\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-21.1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 21.1\" width=\"584\" height=\"495\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-21.1.png 584w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-21.1-300x254.png 300w\" sizes=\"(max-width: 584px) 100vw, 584px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49001\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-21.2.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 21.2\" width=\"545\" height=\"636\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-21.2.png 545w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-21.2-257x300.png 257w\" sizes=\"(max-width: 545px) 100vw, 545px\" \/><\/p>\n<p>Question 22.<br \/>\nT(-2, 1), U( 2, 1), V(0, 4)<\/p>\n<p>Answer:<br \/>\nThe orthocenter lies inside the triangle ABC.<\/p>\n<p>Explanation:<br \/>\nThe slope of UV = \\(\\frac { 4 &#8211; 1 }{ 0 &#8211; 2 } \\) = \\(\\frac { -3 }{ 2 } \\)<br \/>\nThe slope of the perpendicular line is \\(\\frac { 2 }{ 3 } \\)<br \/>\nThe equation of the perpendicular line is (y &#8211; 1) = \\(\\frac { 2 }{ 3 } \\)(x + 2)<br \/>\n3(y &#8211; 1) = 2(x + 2)<br \/>\n3y &#8211; 3 = 2x + 2<br \/>\n2x &#8211; 3y + 5 = 0 &#8212; (i)<br \/>\nThe slope of TV = \\(\\frac { 4 &#8211; 1 }{ 0 + 2 } \\) = \\(\\frac { 3 }{ 2 } \\)<br \/>\nThe slope of the perpendicular line is \\(\\frac { -2 }{ 3 } \\)<br \/>\nThe equation of the perpendicular line is (y &#8211; 1) = \\(\\frac { -2 }{ 3 } \\)(x &#8211; 2)<br \/>\n3(y &#8211; 1) = -2(x &#8211; 2)<br \/>\n3y &#8211; 3 = -2x + 4<br \/>\n2x + 3y &#8211; 7 = 0 &#8212;(ii)<br \/>\nAdd two equations<br \/>\n2x &#8211; 3y + 5 + 2x + 3y &#8211; 7 = 0<br \/>\n4x &#8211; 2 = 0<br \/>\nx = 0.5<br \/>\n2x &#8211; 1.5 + 5 = 0<br \/>\nx = -1.75<br \/>\nSo, the orthocenter is (0, 2.33)<br \/>\nThe orthocenter lies inside the triangle ABC.<\/p>\n<p><strong>CONSTRUCTION<\/strong><br \/>\nIn Exercises 23-26, draw the indicated triangle and find its centroid and orthocenter.<br \/>\nQuestion 23.<br \/>\nisosceles right triangle<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49002\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-23.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 23\" width=\"588\" height=\"336\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-23.png 588w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-23-300x171.png 300w\" sizes=\"(max-width: 588px) 100vw, 588px\" \/><\/p>\n<p>Question 24.<br \/>\nobtuse scalene triangle<\/p>\n<p>Answer:<br \/>\nConstruct the medians for all the sides of the trinagle by finding the midpoint. The point where all medians intersect is centroid. The orthocenter lies outside the triangle for an obtuse triangle.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-106743\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-answers-geometry-chapter-6-relationships-within-triangles-6.3-1.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 6.3 1\" width=\"406\" height=\"245\" \/><\/p>\n<p>Question 25.<br \/>\nright scalene triangle<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49003\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-25.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 25\" width=\"577\" height=\"340\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-25.png 577w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-25-300x177.png 300w\" sizes=\"(max-width: 577px) 100vw, 577px\" \/><\/p>\n<p>Question 26.<br \/>\nacute isosceles triangle<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-106745\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-answers-geometry-chapter-6-relationships-within-triangles-6.3-2.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 6.3 2\" width=\"209\" height=\"276\" \/><\/p>\n<p><strong>ERROR ANALYSIS<\/strong><br \/>\nIn Exercises 27 and 28, describe and correct the error in finding DE. Point D is the centroid of \u2206ABC.<\/p>\n<p>Question 27.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42237\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-82.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 82\" width=\"336\" height=\"125\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-82.png 336w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-82-300x112.png 300w\" sizes=\"(max-width: 336px) 100vw, 336px\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49004\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-27.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 27\" width=\"571\" height=\"176\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-27.png 571w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-27-300x92.png 300w\" sizes=\"(max-width: 571px) 100vw, 571px\" \/><\/p>\n<p>Question 28.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42236\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-83.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 83\" width=\"342\" height=\"124\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-83.png 342w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-83-300x109.png 300w\" sizes=\"(max-width: 342px) 100vw, 342px\" \/><\/p>\n<p>Answer:<br \/>\nThe length of DE should be 1\/3 of length of AE because it is the shorter segment from the centroid<br \/>\nDE = \\(\\frac { 1 }{ 3 } \\)AE<br \/>\n= \\(\\frac { 1 }{ 3 } \\)(24)<br \/>\n= 8<br \/>\nDE = 8<\/p>\n<p><strong>PROOF<\/strong><br \/>\nIn Exercises 29 and 30, write a proof of the statement.<\/p>\n<p>Question 29.<br \/>\nThe angle bisector from the vertex angle to the base of an isosceles triangle is also a median.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49005\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-29.1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 29.1\" width=\"574\" height=\"273\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-29.1.png 574w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-29.1-300x143.png 300w\" sizes=\"(max-width: 574px) 100vw, 574px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49006\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-29.2.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 29.2\" width=\"611\" height=\"603\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-29.2.png 611w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-29.2-300x296.png 300w\" sizes=\"(max-width: 611px) 100vw, 611px\" \/><\/p>\n<p>Question 30.<br \/>\nThe altitude from the vertex angle to the base of an isosceles triangle is also a perpendicular bisector.<br \/>\nAnswer:<\/p>\n<p><strong>CRITICAL THINKING<\/strong><br \/>\nIn Exercises 31-36, complete the statement with always, sometimes, or never. Explain your reasoning.<\/p>\n<p>Question 31.<br \/>\nThe centroid is _____________ on the triangle.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49007\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-31.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 31\" width=\"541\" height=\"112\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-31.png 541w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-31-300x62.png 300w\" sizes=\"(max-width: 541px) 100vw, 541px\" \/><\/p>\n<p>Question 32.<br \/>\nThe orthocenter is _____________ outside the triangle.<\/p>\n<p>Answer:<br \/>\nThe orthocenter is always outside the triangle opposite the longest leg, on the same side as the largest angle. The only time all three of these centers fall in the same spot is in the case of an equilateral triangle. In fact, in this case, the incenter falls in the same place as well.<\/p>\n<p>Question 33.<br \/>\nA median is _____________ the same line segment as a perpendicular bisector.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49008\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-33.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 33\" width=\"554\" height=\"172\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-33.png 554w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-33-300x93.png 300w\" sizes=\"(max-width: 554px) 100vw, 554px\" \/><\/p>\n<p>Question 34.<br \/>\nAn altitude is ______________ the same line segment as an angle bisector.<\/p>\n<p>Answer:<\/p>\n<p>Question 35.<br \/>\nThe centroid and orthocenter are _____________ the same point.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49009\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-35.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 35\" width=\"578\" height=\"87\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-35.png 578w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-35-300x45.png 300w\" sizes=\"(max-width: 578px) 100vw, 578px\" \/><\/p>\n<p>Question 36.<br \/>\nThe centroid is ______________ formed by the intersection o\u00cd the three medians.<\/p>\n<p>Answer:<br \/>\nThe centroid is a point formed by the intersection o\u00cd the three medians.<\/p>\n<p>Question 37.<br \/>\n<strong>WRITING<\/strong><br \/>\nCompare an altitude of a triangle with a perpendicular bisector of a triangle.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49010\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-37.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 37\" width=\"573\" height=\"170\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-37.png 573w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-37-300x89.png 300w\" sizes=\"(max-width: 573px) 100vw, 573px\" \/><\/p>\n<p>Question 38.<br \/>\n<strong>WRITING<\/strong><br \/>\nCompare a median. an altitude, and an angle bisector of a triangle.<\/p>\n<p>Answer:<br \/>\nMedian is a line segment joining a vertex of a triangle with the midoint of the opposite side. Angle bisector is a line segment joining a vertex of a trianglr with the opposite side such that the angle at the vertex split into two equal parts. While altitude is also a line segment joining a vertex of a triangle with opposite side such that the segment is perpendicular to the opposite side. In general, medians, angle bisectors and altitudes drawn from the same vertex of a triangle are different line segments. In an equilateral triagle, altitude, median, and angle bisector drawn from the same vertex overlap. In an isosceles trianglle, the altitude drawn to the base is the median and the angle bisector, the median drawn to the base is the altitude and the angle bisector, the bisector of the angle opposite to the base is the altitude and the median.<\/p>\n<p>Question 39.<br \/>\n<strong>MODELING WITH MATHEMATICS<\/strong><br \/>\nFind the area of the triangular part of the paper airplane wing that is outlined in red. Which special segment of the triangle did you use?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42235\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-84.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 84\" width=\"266\" height=\"202\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49011\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-39.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 39\" width=\"522\" height=\"141\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-39.png 522w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-39-300x81.png 300w\" sizes=\"(max-width: 522px) 100vw, 522px\" \/><\/p>\n<p>Question 40.<br \/>\n<strong>ANALYZING RELATIONSHIPS<\/strong><br \/>\nCopy and complete the statement for \u2206DEF with centroid K and medians<br \/>\n\\(\\overline{D H}\\), \\(\\overline{E J}\\), and \\(\\overline{F G}\\).<\/p>\n<p>a. EJ = ____ KJ<\/p>\n<p>Answer:<br \/>\nEJ = 2KJ<\/p>\n<p>Explanation:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-106749\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-answers-geometry-chapter-6-relationships-within-triangles-6.3-3.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 6.3 3\" width=\"294\" height=\"372\" \/><br \/>\nSince KJ is the shorter distance between the two parts of the median, its length is 1\/3 of the EJ.<\/p>\n<p>b. DK = ____ KH<\/p>\n<p>Answer:<br \/>\nDK = 2 KH<br \/>\nSince KH is the shorter distance between the two parts of the median, its length is 1\/3 of the DH.<\/p>\n<p>c. FG = ___ KF<\/p>\n<p>Answer:<br \/>\nFG = \\(\\frac { 2 }{ 3 } \\)KF<br \/>\nSince KF is the longest distance between two parts, its length is \\(\\frac { 2 }{ 3 } \\) of FG<\/p>\n<p>d. KG = ___ FG<\/p>\n<p>Answer:<br \/>\nKG = 2FG<br \/>\nSince KG is the shorter distance between the two parts of the median, its length is 1\/3 of the FG.<\/p>\n<p><strong>MATHEMATCAL CONNETIONS<\/strong><br \/>\nIn Exercises 41-44, point D is the centroid of \u2206ABC. Use the given information to find the value of x.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-42234\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-85.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 85\" width=\"162\" height=\"121\" \/><\/p>\n<p>Question 41.<br \/>\nBD = 4x + 5 and BF = 9x<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49012\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-41.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 41\" width=\"179\" height=\"187\" \/><\/p>\n<p>Question 42.<br \/>\nGD = 2x &#8211; 8 and GC = 3x + 3<\/p>\n<p>Answer:<br \/>\nx = 9<\/p>\n<p>Explanation:<br \/>\nGD = \\(\\frac { 1 }{ 3 } \\)GC<br \/>\n2x &#8211; 8 = \\(\\frac { 1 }{ 3 } \\)(3x + 3)<br \/>\n2x &#8211; 8 = x + 1<br \/>\nx = 9<\/p>\n<p>Question 43.<br \/>\nAD = 5x and DE = 3x &#8211; 2<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49013\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-43.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 43\" width=\"174\" height=\"174\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-43.png 174w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-43-150x150.png 150w\" sizes=\"(max-width: 174px) 100vw, 174px\" \/><\/p>\n<p>Question 44.<br \/>\nDF = 4x &#8211; 1 and BD = 6x + 4<\/p>\n<p>Answer:<br \/>\nx = 3<\/p>\n<p>Explanation:<br \/>\nBD = 2DF<br \/>\n6x + 4 = 2(4x &#8211; 1)<br \/>\n6x + 4 = 8x &#8211; 2<br \/>\n2x = 6<br \/>\nx = 3<\/p>\n<p>Question 45.<br \/>\n<strong>MATHEMATICAL CONNECTIONS<\/strong><br \/>\nGraph the lines on the same coordinate plane. Find the centroid of the triangle formed by their intersections.<br \/>\ny<sub>1<\/sub> = 3x &#8211; 4<br \/>\ny<sub>2<\/sub> = \\(\\frac{3}{4}\\)x + 5<br \/>\ny<sub>2<\/sub> = &#8211; \\(\\frac{3}{2}\\)x &#8211; 4<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49014\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-45.1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 45.1\" width=\"438\" height=\"410\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-45.1.png 438w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-45.1-300x281.png 300w\" sizes=\"(max-width: 438px) 100vw, 438px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49015\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-45.2.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 45.2\" width=\"577\" height=\"336\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-45.2.png 577w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-45.2-300x175.png 300w\" sizes=\"(max-width: 577px) 100vw, 577px\" \/><\/p>\n<p>Question 46.<br \/>\n<strong>CRITICAL THINKING<\/strong><br \/>\nIn what types of triangles can a vertex be one of the points of concurrency of the triangle? Explain your reasoning.<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-106754\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-answers-geometry-chapter-6-relationships-within-triangles-6.3-4.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 6.3 4\" width=\"282\" height=\"274\" \/><br \/>\nIn right angles, the orthocenter forms at the 90 degrees angle.<\/p>\n<p>Question 47.<br \/>\n<strong>WRITING EQUATIONS<\/strong><br \/>\nUse the numbers and symbols to write three different equations for PE.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42233\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-86.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 86\" width=\"189\" height=\"133\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42232\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-87.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 87\" width=\"352\" height=\"150\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-87.png 352w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-87-300x128.png 300w\" sizes=\"(max-width: 352px) 100vw, 352px\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49016\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-47.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 47\" width=\"371\" height=\"46\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-47.png 371w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-47-300x37.png 300w\" sizes=\"(max-width: 371px) 100vw, 371px\" \/><\/p>\n<p>Question 48.<br \/>\n<strong>HOW DO YOU SEE IT?<\/strong><br \/>\nUse the figure.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42231\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-88.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 88\" width=\"222\" height=\"150\" \/><br \/>\na. What type of segment is \\(\\overline{K M}\\)? Which point of concurrency lies on \\(\\overline{K M}\\)?<\/p>\n<p>Answer:<br \/>\nKM is the median drawn from K to JL.<\/p>\n<p>b. What type of segments is \\(\\overline{K N}\\)? Which point of concurrency lies on \\(\\overline{K N}\\)?<\/p>\n<p>Answer:<br \/>\nKN is the altitude drawn from K to JL.<\/p>\n<p>c. Compare the areas of \u2206JKM and \u2206KLM. Do you think the areas of the triangles formed by the median of any triangle will always compare this way? Explain your reasoning.<\/p>\n<p>Answer:<\/p>\n<p>Question 49.<br \/>\n<strong>MAKING AN ARGUMENT<\/strong><br \/>\nYour friend claims that it is possible for the circumcenter, incenter, centroid, and orthocenter to all be the same point. Do you agree? Explain your reasoning.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49017\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-49.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 49\" width=\"581\" height=\"87\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-49.png 581w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-49-300x45.png 300w\" sizes=\"(max-width: 581px) 100vw, 581px\" \/><\/p>\n<p>Question 50.<br \/>\n<strong>DRAWING CONCLUSIONS<\/strong><br \/>\nThe center of gravity of<br \/>\na triangle, the point where a triangle can balance on the tip of a pencil, is one of the four points of concurrency. Draw and cut out a large scalene triangle on a piece of cardboard. Which of the four points of concurrency is the center of gravity? Explain.<\/p>\n<p>Answer:<br \/>\nThe median is the balancing line of the triangle<br \/>\nThe centroid is the point of concurrency of three medians<br \/>\nThe centroid is the center of gravity of the triangle.<\/p>\n<p>Explanation:<br \/>\nThe median is the balancing line of the triangle i.e median divides the triangle into two triangles of equal areas.<br \/>\nAssuming trinagle ABC<br \/>\nThe area of ABC = 0.5bh<br \/>\nThe median divides base into two equal parts, each forming a smaller triangle with the same height of the original one.<br \/>\nAD is the median of base BC<br \/>\nSo, bD = CD = \\(\\frac { 1 }{ 2 } \\) BC<br \/>\nThe area of ABC = \\(\\frac { 1 }{ 2 } \\) x BC x h<br \/>\nArea of ABD = \\(\\frac { 1 }{ 2 } \\) x BD x h = \\(\\frac { 1 }{ 2 } \\) x ABC area<br \/>\nArea of ACD = \\(\\frac { 1 }{ 2 } \\) x CD x h = \\(\\frac { 1 }{ 2 } \\) x ABC area<br \/>\nThe centroid is the concurrency point of three medians.<br \/>\nSo, the centroid is the gravity centre of the triangle.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-106762\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-answers-geometry-chapter-6-relationships-within-triangles-6.3-5.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 6.3 5\" width=\"1000\" height=\"896\" \/><\/p>\n<p>Question 51.<br \/>\n<strong>PROOF<\/strong><br \/>\nProse that a median of an equilateral triangle is also an angle bisector, perpendicular bisector, and altitude.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49018\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-51.1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 51.1\" width=\"532\" height=\"185\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-51.1.png 532w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-51.1-300x104.png 300w\" sizes=\"(max-width: 532px) 100vw, 532px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49019\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-51.2.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 51.2\" width=\"558\" height=\"494\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-51.2.png 558w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-51.2-300x266.png 300w\" sizes=\"(max-width: 558px) 100vw, 558px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49020\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-51.3.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 51.3\" width=\"561\" height=\"485\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-51.3.png 561w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-51.3-300x259.png 300w\" sizes=\"(max-width: 561px) 100vw, 561px\" \/><\/p>\n<p>Question 52.<br \/>\n<strong>THOUGHT PROVOKING<\/strong><br \/>\nConstruct an acute scalene triangle. Find the orthocenter, centroid, and circumcenter. What can you conclude about the three points of concurrency?<\/p>\n<p>Answer:<br \/>\nThe centroid, orthocenter and circumcenter are collinear.<\/p>\n<p>Explanation:<br \/>\nBelow is the scalene triangle with the centroid, orthocenter and circumcenter<br \/>\nPoint G is the centroid, H is the orthocenter and J is the circumcenter.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-106765\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-answers-geometry-chapter-6-relationships-within-triangles-6.3-6.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 6.3 6\" width=\"2288\" height=\"1441\" \/><br \/>\nIf we draw a line through these points, we can see all the points lie on the same line.<br \/>\nTherefore, we can conclude that the centroid, orthocenter and circumcenter are collinear.<br \/>\nBelow mentioned is the construction part.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-106764\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-answers-geometry-chapter-6-relationships-within-triangles-6.3-7.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 6.3 6\" width=\"2237\" height=\"1369\" \/><\/p>\n<p>Question 53.<br \/>\n<strong>CONSTRUCTION<\/strong><br \/>\nFollow the steps to construct a nine-point circle. Why is it called a nine-point circle?<br \/>\nStep 1 Construct a large acute scalene triangle.<br \/>\nStep 2 Find the orthocenter and circumcenter of the triangle.<br \/>\nStep 3 Find the midpoint between the orthocenter and circumcenter.<br \/>\nStep 4 Find the midpoint between each vertex and the orthocenter.<br \/>\nStep 5 Construct a circle. Use the midpoint in Step 3 as the center of the circle, and the distance from the center to the midpoint of a side of the triangle as the radius.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49021\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-53.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 53\" width=\"593\" height=\"630\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-53.png 593w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-53-282x300.png 282w\" sizes=\"(max-width: 593px) 100vw, 593px\" \/><\/p>\n<p>Question 54.<br \/>\n<strong>PROOF<\/strong><br \/>\nProve the statements in parts (a)-(c).<br \/>\nGiven \\(\\overline{L P}\\) and \\(\\overline{M Q}\\) are medians of scalenc \u2206LMN.<br \/>\nPoint R is on \\(\\vec{L}\\)P such that \\(\\overline{L P} \\cong \\overline{P R}\\). Point S is on \\(\\vec{M}\\)Q such that \\(\\overline{M Q} \\cong \\overline{Q S}\\).<br \/>\nProve<br \/>\na. \\(\\overline{N S} \\cong \\overline{N R}\\)<br \/>\nb. \\(\\overline{N S}\\) and \\(\\overline{N R}\\) are both parallel to \\(\\overline{L M}\\).<br \/>\nc. R, N, and S are collinear.<br \/>\nAnswer:<\/p>\n<p>Maintaining Mathematical Proficiency<\/p>\n<p>Determine whether \\(\\overline{A B}\\) is parallel to \\(\\overline{C D}\\).<\/p>\n<p>Question 55.<br \/>\nA(5, 6), B (- 1, 3), C(- 4, 9), D(- 16, 3)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49022\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-55.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 55\" width=\"502\" height=\"150\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-55.png 502w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-55-300x90.png 300w\" sizes=\"(max-width: 502px) 100vw, 502px\" \/><\/p>\n<p>Question 56.<br \/>\nA(- 3, 6), B(5, 4), C(- 14, &#8211; 10), D(- 2, &#8211; 7)<\/p>\n<p>Answer:<br \/>\nAB is not parallel to CD.<\/p>\n<p>Explanation:<br \/>\nSlope of AB = \\(\\frac { 4 &#8211; 6 }{ 5 + 3 } \\) = \\(\\frac { -1 }{ 4 } \\)<br \/>\nSlopeof CD = \\(\\frac { -7 + 10 }{ -2 + 14 } \\) = \\(\\frac { 1 }{ 4 } \\)<br \/>\nThe slopes of AB and CD are not equal. So AB and CD are not parallel.<\/p>\n<p>Question 57.<br \/>\nA (6, &#8211; 3), B(5, 2), C(- 4, &#8211; 4), D(- 5, 2)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49023\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-57.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.3 Question 57\" width=\"575\" height=\"175\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-57.png 575w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.3-Question-57-300x91.png 300w\" sizes=\"(max-width: 575px) 100vw, 575px\" \/><\/p>\n<p>Question 58.<br \/>\nA(- 5, 6), B(- 7, 2), C(7, 1), D(4, &#8211; 5)<\/p>\n<p>Answer:<br \/>\nAB is parallel to CD.<\/p>\n<p>Explanation:<br \/>\nSlope of AB = \\(\\frac { 2 &#8211; 6 }{ -7 + 5 } \\) = 2<br \/>\nSlopeof CD = \\(\\frac { -5 &#8211; 1 }{ 4 &#8211; 7 } \\) = 2<br \/>\nThe slopes of AB and CD are equal. So, AB is parallel to CD.<\/p>\n<h3><a id=\"6.1_and_6.3_Quiz\"><\/a>6.1 and 6.3 Quiz<\/h3>\n<p>Find the indicated measure. Explain your reasoning.<\/p>\n<p>Question 1.<br \/>\nUV<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42266\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-89.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 89\" width=\"163\" height=\"130\" \/><\/p>\n<p>Answer:<br \/>\nUV = 15<\/p>\n<p>Explanation:<br \/>\nSV = VU<br \/>\n2x + 11 = 8x &#8211; 1<br \/>\n8x &#8211; 2x = 11 + 1<br \/>\n6x = 12<br \/>\nx = 2<br \/>\nUV = 8(2) &#8211; 1 = 15<\/p>\n<p>Question 2.<br \/>\nQP<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42265\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-90.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 90\" width=\"175\" height=\"119\" \/><\/p>\n<p>Answer:<br \/>\nQP = 18<\/p>\n<p>Explanation:<br \/>\nQP = QR<br \/>\n6x = 3x + 9<br \/>\n3x = 9<br \/>\nx = 3<br \/>\nQP = 6(3) = 18<\/p>\n<p>Question 3.<br \/>\nm\u2220GJK<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42264\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-91.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 91\" width=\"148\" height=\"138\" \/><\/p>\n<p>Answer:<br \/>\nm\u2220GJK = 59\u00b0<\/p>\n<p>Explanation:<br \/>\n5x &#8211; 4 = 4x + 3<br \/>\nx = 7<br \/>\n\u2220JGK = 4(7) + 3 = 31<br \/>\nm\u2220GJK = 180 &#8211; (31 + 90) = 180 &#8211; 121 = 59<\/p>\n<p>Find the coordinates of the circumcenter of the triangle with the given vertices.<\/p>\n<p>Question 4.<br \/>\nA(- 4, 2), B(- 4, &#8211; 4), C(0, &#8211; 4)<\/p>\n<p>Answer:<br \/>\nThe circumcenter is (-2, -1)<\/p>\n<p>Explanation:<br \/>\nRecall that the circumcentre of a triangle is equidistant from the vertices of a triangle. Let A(- 4, 2), B(- 4, &#8211; 4), C(0, &#8211; 4) be the vertices of the given triangle and let P(x,y) be the circumcentre of this triangle. Then<br \/>\nPA = PB = PC<br \/>\nPA\u00b2 = PB\u00b2 = PC\u00b2<br \/>\nPA\u00b2 = PB\u00b2<br \/>\n(x + 4)\u00b2 + (y &#8211; 2)\u00b2 = (x + 4)\u00b2 + (y + 4)\u00b2<br \/>\nx\u00b2 + 8x + 16 + y\u00b2 &#8211; 4y + 4 = x\u00b2 + 8x + 16 + y\u00b2 + 8y + 16<br \/>\n12y = -12<br \/>\ny = -1<br \/>\nPB\u00b2 = PC\u00b2<br \/>\n(x + 4)\u00b2 + (y + 4)\u00b2 = (x &#8211; 0)\u00b2 + (y + 4)\u00b2<br \/>\nx\u00b2 + 8x + 16 + y\u00b2 + 8y + 16 = x\u00b2 + y\u00b2 + 8y + 16<br \/>\n8x = -16<br \/>\nx = -2<br \/>\nThe circumcenter is (-2, -1)<\/p>\n<p>Question 5.<br \/>\nD(3, 5), E(7, 9), F(11, 5)<\/p>\n<p>Answer:<br \/>\nThe circumcenter is (7, 5)<\/p>\n<p>Explanation:<br \/>\nRecall that the circumcentre of a triangle is equidistant from the vertices of a triangle. Let D(3, 5), E(7, 9), F(11, 5) be the vertices of the given triangle and let P(x,y) be the circumcentre of this triangle. Then<br \/>\nPD = PE = PF<br \/>\nPD\u00b2 = PE\u00b2 = PF\u00b2<br \/>\nPD\u00b2 = PE\u00b2<br \/>\n(x &#8211; 3)\u00b2 + (y &#8211; 5)\u00b2 = (x &#8211; 7)\u00b2 + (y &#8211; 9)\u00b2<br \/>\nx\u00b2 &#8211; 6x + 9 + y\u00b2 &#8211; 10y + 25 = x\u00b2 &#8211; 14x + 49 + y\u00b2 &#8211; 18y + 81<br \/>\n-6x + 14x &#8211; 10y + 18y = 130 &#8211; 34<br \/>\n8x + 8y = 96<br \/>\nx + y = 12 &#8212; (i)<br \/>\nPE\u00b2 = PF\u00b2<br \/>\n(x &#8211; 7)\u00b2 + (y &#8211; 9)\u00b2 = (x &#8211; 11)\u00b2 + (y &#8211; 5)\u00b2<br \/>\nx\u00b2 &#8211; 14x + 49 + y\u00b2 &#8211; 18y + 81 = x\u00b2 &#8211; 22x + 121 + y\u00b2 &#8211; 10y + 25<br \/>\n-14x + 22x &#8211; 18y + 10y = 146 &#8211; 130<br \/>\n8x &#8211; 8y = 16<br \/>\nx &#8211; y = 2 &#8212; (ii)<br \/>\nAdd (i) &amp; (ii)<br \/>\nx + y + x &#8211; y = 12 + 2<br \/>\n2x = 14<br \/>\nx = 7<br \/>\nPut x = 7 in (i)<br \/>\n7 + y = 12<br \/>\ny = 5<br \/>\nThe circumcenter is (7, 5)<\/p>\n<p>The incenter of \u2206ABC is point N. Use the given information to find the indicated measure.<\/p>\n<p>Question 6.<br \/>\nNQ = 2x + 1, NR = 4x &#8211; 9<br \/>\nFind NS.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42263\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-92.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 92\" width=\"229\" height=\"123\" \/><\/p>\n<p>Answer:<br \/>\nNS = 11<\/p>\n<p>Explanation:<br \/>\nNQ = NR = NS<br \/>\n2x + 1 = 4x &#8211; 9<br \/>\n4x &#8211; 2x = 10<br \/>\n2x = 10<br \/>\nx = 5<br \/>\nNQ = 10 + 1 = 11<br \/>\nNS = 11<\/p>\n<p>Question 7.<br \/>\nNU = &#8211; 3x + 6, NV = &#8211; 5x<br \/>\nFind NT.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42262\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-93.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 93\" width=\"216\" height=\"115\" \/><\/p>\n<p>Answer:<br \/>\nNT = 15<\/p>\n<p>Explanation:<br \/>\nNU = NV = NT<br \/>\n-3x + 6 = -5x<br \/>\n-3x + 5x = -6<br \/>\n2x = -6<br \/>\nx = -3<br \/>\nNT = -5(-3) = 15<\/p>\n<p>Question 8.<br \/>\nNZ = 4x &#8211; 10, NY = 3x &#8211; 1<br \/>\nFind NW.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-42261\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-94.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 94\" width=\"225\" height=\"131\" \/><\/p>\n<p>Answer:<br \/>\nNW = 26<\/p>\n<p>Explanation:<br \/>\nNZ = NY = NW<br \/>\n4x &#8211; 10 = 3x &#8211; 1<br \/>\nx = 9<br \/>\nNZ = 4(9) &#8211; 10 = 36 &#8211; 10 = 26<br \/>\nNW = 26<\/p>\n<p>Find the coordinates of the centroid of the triangle wilt the given vertices.<br \/>\nQuestion 9.<br \/>\nJ(- 1, 2), K(5, 6), L(5, &#8211; 2)<\/p>\n<p>Answer:<br \/>\nCentroid of the triangle is (3, 2)<\/p>\n<p>Explanation:<br \/>\nCentroid of the triangle = (\\(\\frac { -1 + 5 + 5 }{ 3 } \\), \\(\\frac { 2 + 6 &#8211; 2 }{ 3 } \\))<br \/>\n= (3, 2)<\/p>\n<p>Question 10.<br \/>\nM(- 8, &#8211; 6), N(- 4, &#8211; 2), P(0, &#8211; 4)<\/p>\n<p>Answer:<br \/>\nCentroid of the triangle is (-4, -4)<\/p>\n<p>Explanation:<br \/>\nCentroid of the triangle = (\\(\\frac { -8 &#8211; 4 + 0 }{ 3 } \\), \\(\\frac { -6 &#8211; 2 &#8211; 4 }{ 3 } \\))<br \/>\n= (-4, -4)<\/p>\n<p>Tell whether the orthocenter is inside, on, or outside the triangle. Then find its coordinates.<\/p>\n<p>Question 11.<br \/>\nT(- 2, 5), U(0, 1), V(2, 5)<\/p>\n<p>Answer:<br \/>\nThe orthocenter lies inside the triangle TUV<\/p>\n<p>Explanation:<br \/>\nThe slope of TU = \\(\\frac { 1 &#8211; 5 }{ 0 + 2 } \\) = -2<br \/>\nThe slope of the perpendicular line is \\(\\frac { 1 }{ 2 } \\)<br \/>\nThe perpendicular line is y &#8211; 5 = \\(\\frac { 1 }{ 2 } \\)(x &#8211; 2)<br \/>\n2y &#8211; 10 = x &#8211; 2<br \/>\nx &#8211; 2y + 8 = 0<br \/>\nThe slope of UV = \\(\\frac { 5 &#8211; 1 }{ 2 &#8211; 0 } \\) = 2<br \/>\nThe slope of the perpendicular line is \\(\\frac { -1 }{ 2 } \\)<br \/>\nThe perpendicular line is y &#8211; 5 = \\(\\frac { -1 }{ 2 } \\)(x + 2)<br \/>\n2y &#8211; 10 = -x &#8211; 2<br \/>\nx + 2y &#8211; 8 = 0<br \/>\nequate both equations<br \/>\nx &#8211; 2y + 8 = x + 2y &#8211; 8<br \/>\n-4y = -16<br \/>\ny = 4<br \/>\nx &#8211; 2(4) + 8 = 0<br \/>\nx = 0<br \/>\nSo, the orthocenter is (0, 4)<br \/>\nThe orthocenter lies inside the triangle TUV<\/p>\n<p>Question 12.<br \/>\nX(- 1, &#8211; 4), Y(7, &#8211; 4), Z(7, 4)<\/p>\n<p>Answer:<br \/>\nThe orthocenter lies on the vertex.<\/p>\n<p>Explanation:<br \/>\nThe slope of XY = \\(\\frac { -4 + 4 }{ 7 + 1 } \\) = 0<br \/>\nThe slope of the perpendicular line is 0<br \/>\nThe perpendicular line is y &#8211; 4 = 0<br \/>\ny = 4<br \/>\nThe slope of XZ =\\(\\frac { 4 + 4 }{ 7 + 1} \\) = 1<br \/>\nThe slope of the perpendicular line is -1<br \/>\nThe perpendicular line is y + 4 = -1(x &#8211; 7)<br \/>\ny + 4 = -x + 7<br \/>\nx + y &#8211; 3 = 0<br \/>\nx + 4 &#8211; 3 = 0<br \/>\nx = -1<br \/>\nSo, the orthocenter is (-1, 4)<br \/>\nThe orthocenter lies on the vertex.<\/p>\n<p>Question 13.<br \/>\nA woodworker is culling the largest wheel possible from a triangular scrap of wood. The wheel just touches each side of the triangle, as shown.<br \/>\nAnswer:<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-42260\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-95.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 95\" width=\"265\" height=\"155\" \/><\/p>\n<p>a. Which point of concurrency is the center of the circle? What type<br \/>\nof segments are \\(\\overline{B G}\\), \\(\\overline{C G}\\), and \\(\\overline{A G}\\)?<br \/>\nAnswer:<\/p>\n<p>b.<br \/>\nWhich theorem can you use to prove that \u2206BGF \u2245 \u2206BGE?<br \/>\nAnswer:<\/p>\n<p>c. Find the radius of the wheel to the nearest tenth of a centimeter. Justify your answer.<\/p>\n<p>Answer:<\/p>\n<p>Question 14.<br \/>\nThe Deer County Parks Committee plans to build a park at point P, equidistant from the three largest cities labeled X, Y, and Z. The map shown was created b the committee.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-42258\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-96.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 96\" width=\"309\" height=\"193\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-96.png 309w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-96-300x187.png 300w\" sizes=\"(max-width: 309px) 100vw, 309px\" \/><\/p>\n<p>a. Which point of concurrency did the commIttee use as the location of the Park?<br \/>\nAnswer:<\/p>\n<p>b. Did the committee use the best point of concurrency for the location of the park? Ii not, which point would be better to use? Explain.<br \/>\nAnswer:<\/p>\n<h3><a id=\"6.4_The_Triangle_Midsegment_Theorem\"><\/a>6.4 The Triangle Midsegment Theorem<\/h3>\n<p><strong>Exploration 1<\/strong><\/p>\n<p>Midsegments of a Triangle<\/p>\n<p>Work with a partner. Use dynamic geometry software. Draw any \u2206ABC.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-43046\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-97.png\" alt=\"Big Ideas Math Geometry Solutions Chapter 6 Relationships Within Triangles 97\" width=\"417\" height=\"246\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-97.png 417w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-97-300x177.png 300w\" sizes=\"(max-width: 417px) 100vw, 417px\" \/><\/p>\n<p>a. Plot midpoint D of \\(\\overline{A B}\\) and midpoint E of \\(\\overline{B C}\\). Draw \\(\\overline{D E}\\), which is a midsegment of \u2206ABC.<br \/>\nAnswer:<\/p>\n<p>b. Compare the slope and length of \\(\\overline{D E}\\) with the slope and length of \\(\\overline{A C}\\).<br \/>\nAnswer:<\/p>\n<p>c. Write a conjecture about the relationships between the midsegments and sides of a triangle. Test your conjecture by drawing the other midsegments of \u2206ABC, dragging vertices to change \u2206ABC. and noting whether the relationships hold.<br \/>\nAnswer:<\/p>\n<p><strong>Exploration 2<\/strong><\/p>\n<p>Midsegments of a Triangle<\/p>\n<p>Work with a partner. Use dynamic geometry software. Draw any \u2206ABC.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-43045\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-98.png\" alt=\"Big Ideas Math Geometry Solutions Chapter 6 Relationships Within Triangles 98\" width=\"515\" height=\"245\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-98.png 515w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-98-300x143.png 300w\" sizes=\"(max-width: 515px) 100vw, 515px\" \/><\/p>\n<p>a. Draw all three midsegments of \u2206ABC.<br \/>\nAnswer:<\/p>\n<p>b. Use the drawing to write a Conjecture about the triangle formed by the midsegments of the original triangle.<br \/>\n<strong>CONSTRUCTING VIABLE ARGUMENTS<\/strong><br \/>\nTo be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures.<br \/>\nAnswer:<\/p>\n<p>Communicate Your Answer<\/p>\n<p>Question 3.<br \/>\nHow are the midsegments of a triangle related to the sides of the triangle?<br \/>\nAnswer:<\/p>\n<p>Question 4.<br \/>\nIn \u2206RST. \\(\\overline{U V}\\) is the rnidsegrnent connecting the midpoints of \\(\\overline{R S}\\) and \\(\\overline{S T}\\). Given<br \/>\nUV = 12, find RT.<br \/>\nAnswer:<\/p>\n<h3><a id=\"Lesson_6.4_The_Triangle_Midsegment_Theorem\"><\/a>Lesson 6.4 The Triangle Midsegment Theorem<\/h3>\n<p><strong>Monitoring progress<\/strong><\/p>\n<p>Use the graph of \u25b3ABC.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-43044\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-99.png\" alt=\"Big Ideas Math Geometry Solutions Chapter 6 Relationships Within Triangles 99\" width=\"203\" height=\"245\" \/><\/p>\n<p>Question 1.<br \/>\nIn \u25b3ABC, show that midsegments \\(\\overline{D E}\\) is parallel to \\(\\overline{A C}\\) and that DE = \\(\\frac { 1 }{ 2 }\\)AC.<\/p>\n<p>Answer:<br \/>\nFind the coordinates of D and E by getting the midpoints of AB, BC<br \/>\nAB midpoint = D(\\(\\frac { 1 &#8211; 1 }{ 2 } \\), \\(\\frac { 4 &#8211; 6 }{ 2 } \\)) = D(0, -1)<br \/>\nMidpoint of BC = E(\\(\\frac { 5 &#8211; 1 }{ 2 } \\), \\(\\frac { 4 + 0 }{ 2 } \\)) = E(2, 2)<br \/>\nSlope of AC = \\(\\frac { 0 + 6 }{ 5 &#8211; 1 } \\) =\\(\\frac { 3 }{ 2 } \\)<br \/>\nSlope of DE = \\(\\frac { 2 + 1 }{ 2 &#8211; 0 } \\) = \\(\\frac { 3 }{ 2 } \\)<br \/>\nThe slopes are same, so the lines are not parallel.<br \/>\nDE = \u221a(2 &#8211; 0)\u00b2 + (2 + 1)\u00b2 = \u221a13<br \/>\nAC = \u221a(5 &#8211; 1)\u00b2 + (0 + 6)\u00b2 = \u221a16 + 36 = \u221a52<br \/>\nSo, DE = \\(\\frac { 1 }{ 2 }\\)AC<\/p>\n<p>Question 2.<br \/>\nFind the coordinates of the endpoints of midsegments \\(\\overline{E F}\\), Which opposite \\(\\overline{A B}\\). show that \\(\\overline{E F}\\) is parallel to \\(\\overline{A B}\\) and that EF = \\(\\frac { 1 }{ 2 }\\)AB.<\/p>\n<p>Answer:<br \/>\nThe coordinates of E = (\\(\\frac { -1 + 5 }{ 2 } \\), \\(\\frac { 0 + 4 }{ 2 } \\)) = (2, 2)<br \/>\nThe coordinates of F = (\\(\\frac { 1 + 5 }{ 2 } \\), \\(\\frac { 0 + 6 }{ 2 } \\)) = (3, 3)<br \/>\nThe slope of AB = \\(\\frac { 4 + 6 }{ -1 &#8211; 1 } \\) = -5<br \/>\nThe slope of EF = \\(\\frac { 3 &#8211; 2 }{ 3 &#8211; 2 } \\) = 1<br \/>\nThe slopes are different. So the lines are not parallel<br \/>\nEF = \u221a(3 &#8211; 2)\u00b2 + (3 &#8211; 2)\u00b2 = \u221a2<br \/>\nAB = \u221a(-1 &#8211; 1)\u00b2 + (4 + 6)\u00b2 = \u221a4 + 100 = \u221a104<\/p>\n<p>Question 3.<br \/>\nIn Example 2, find the coordinates of F, the midpoint of \\(\\overline{O C}\\). Show that \\(\\overline{F E}\\) || \\(\\overline{O B}\\) and FE = \\(\\frac { 1 }{ 2 }\\)OB.<\/p>\n<p>Answer:<br \/>\nThe coordinates of F = (\\(\\frac { 0 + 2p }{ 2 } \\), \\(\\frac { 0 + 0 }{ 2 } \\))<br \/>\n= (p, 0)<br \/>\nThe slope of OB = \\(\\frac { 2r &#8211; 0 }{ 2q &#8211; 0 } \\) = \\(\\frac { r }{ q } \\)<br \/>\nThe slope of FE = \\(\\frac { r &#8211; 0 }{ q + p &#8211; p }\\) = \\(\\frac { r }{ q }\\)<br \/>\nThe slopes are equal. So, the lines are parallel.<br \/>\nFE = \u221a(q + p &#8211; p)\u00b2 + (r &#8211; 0)\u00b2 = \u221aq\u00b2 + r\u00b2<br \/>\nOB = \u221a(2q &#8211; 0)\u00b2 + (2r &#8211; 0)\u00b2 = \u221a4q\u00b2 + 4r\u00b2\u00a0 = 2\u221aq\u00b2 + r\u00b2<br \/>\nOB = 2(FE)<br \/>\nFE = \\(\\frac { 1 }{ 2 }\\)OB.<\/p>\n<p>Question 4.<br \/>\nCopy the diagram in Example 3. Draw and name the third midsegment.<br \/>\nThen find the length of \\(\\overline{V S}\\) when the length of the third midsegment is 81 inches.<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-106909\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-answers-geometry-chapter-6-relationships-within-triangles-6.4-1.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 6.4 1\" width=\"180\" height=\"188\" \/><br \/>\nThe length of third midsegment = 81 in<br \/>\nThe length of VS = \\(\\frac { 81 }{ 2 } \\) = 40.5<\/p>\n<p>Question 5.<br \/>\nIn Example 4. if F is the midpoint of \\(\\overline{C B}\\), what do you know about \\(\\overline{D F}\\)?<\/p>\n<p>Answer:<br \/>\nDF is parallel to AE.<\/p>\n<p>Question 6.<br \/>\n<strong>WHAT IF?<\/strong><br \/>\nIn Example 5, you jog down Peach Street to Plum Street, over Plum Street to Cherry Street. up Cherry Street to Pear Street. over Pear Street to Peach Street. and then back home up Peach Street. Do you jog more miles in Example 5? Explain.<\/p>\n<p>Answer:<br \/>\nThe distance you jog = 2.25 + 1.4 + 1.3 + .65 = 5.6<\/p>\n<h3><a id=\"Exercise_6.4_The_Triangle_Midsegment_Theorem\"><\/a>Exercise 6.4 The Triangle Midsegment Theorem<\/h3>\n<p>Vocabulary and Core Concept Check<\/p>\n<p>Question 1.<br \/>\n<strong>VOCABULARY<\/strong><br \/>\nThe ___________ of a triangle is a segment that connects the midpoints of two sides of the triangle.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49024\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 1\" width=\"556\" height=\"63\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-1.png 556w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-1-300x34.png 300w\" sizes=\"(max-width: 556px) 100vw, 556px\" \/><\/p>\n<p>Question 2.<br \/>\n<strong>COMPLETE THE SENTENCE<br \/>\n<\/strong>If \\(\\overline{D E}\\) is the midsegment opposile \\(\\overline{A C}\\) in \u2206ABC, then \\(\\overline{D E}\\) || \\(\\overline{A C}\\) and DE = ________ AC by the Triangle Midsegrnent Theorem<\/p>\n<p>Answer:<br \/>\nIf \\(\\overline{D E}\\) is the midsegment opposile \\(\\overline{A C}\\) in \u2206ABC, then \\(\\overline{D E}\\) || \\(\\overline{A C}\\) and DE = \\(\\frac { 1 }{ 2 } \\) AC by the Triangle Midsegrnent Theorem (Theorem 6.8).<\/p>\n<p>Monitoring Progress and Modeling with Mathematics<\/p>\n<p>In Exercises 3-6, use the graph of \u2206ABC with midsegments \\(\\overline{D E}\\), \\(\\overline{E F}\\), and \\(\\overline{D F}\\).<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-43043\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-100.png\" alt=\"Big Ideas Math Geometry Solutions Chapter 6 Relationships Within Triangles 100\" width=\"204\" height=\"208\" \/><\/p>\n<p>Question 3.<br \/>\nFind the coordinates of points D, E, and F.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49025\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-3.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 3\" width=\"545\" height=\"30\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-3.png 545w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-3-300x17.png 300w\" sizes=\"(max-width: 545px) 100vw, 545px\" \/><\/p>\n<p>Question 4.<br \/>\nShow that \\(\\overline{D E}\\) is parallel to \\(\\overline{C B}\\) and that DE = \\(\\frac{1}{2}\\)CB.<\/p>\n<p>Answer:<br \/>\nThe slope of DE = \\(\\frac { 0 + 2 }{ -2 + 4 } \\) = 1<br \/>\nThe slope of CB = \\(\\frac { -2 + 6 }{ 1 + 3 } \\)\u00a0 = 1<br \/>\nThe slopes are same. So \\(\\overline{D E}\\) is parallel to \\(\\overline{C B}\\)<br \/>\nDE = \u221a(-2 + 4)\u00b2 + (0 + 2)\u00b2 = \u221a4 + 4 = \u221a8<br \/>\nCB = \u221a(1 + 3)\u00b2 + (-2 + 6)\u00b2 = \u221a16 + 16 = \u221a32<br \/>\nSo, DE = \\(\\frac{1}{2}\\)CB<\/p>\n<p>Question 5.<br \/>\nShow that \\(\\overline{E F}\\) is parallel to \\(\\overline{A C}\\) and that EF = \\(\\frac{1}{2}\\)AC.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49026\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-5.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 5\" width=\"555\" height=\"464\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-5.png 555w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-5-300x251.png 300w\" sizes=\"(max-width: 555px) 100vw, 555px\" \/><\/p>\n<p>Question 6.<br \/>\nShow that \\(\\overline{D F}\\) is parallel to \\(\\overline{A B}\\) and that DF = \\(\\frac{1}{2}\\)AB.<\/p>\n<p>Answer:<br \/>\nThe slope of DF = \\(\\frac { -4 + 2 }{ -1 + 4 } \\) = \\(\\frac { -2 }{ 3 } \\)<br \/>\nThe slope of AB = \\(\\frac { -2 &#8211; 2 }{ 1 + 5 } \\) = \\(\\frac { -4 }{ 6 } \\) = \\(\\frac { -2 }{ 3 } \\)<br \/>\nThe slopes are equal. So, the lines are parallel.<br \/>\nDF = \u221a(-1 + 4)\u00b2 + (-4 + 2)\u00b2 = \u221a9 + 4 = \u221a13<br \/>\nAB = \u221a(1 + 5)\u00b2 + (-2 &#8211; 2)\u00b2 = \u221a36 + 16 = \u221a52<br \/>\nSo, DF = \\(\\frac{1}{2}\\)AB.<\/p>\n<p>In Exercises 7-10, \\(\\overline{D E}\\) is a midsegment of \u2206ABC Find the value of x.<\/p>\n<p>Question 7.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43042\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-101.png\" alt=\"Big Ideas Math Geometry Solutions Chapter 6 Relationships Within Triangles 101\" width=\"174\" height=\"91\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49027\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-7.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 7\" width=\"126\" height=\"110\" \/><\/p>\n<p>Question 8.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43041\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-102.png\" alt=\"Big Ideas Math Geometry Solutions Chapter 6 Relationships Within Triangles 102\" width=\"179\" height=\"91\" \/><\/p>\n<p>Answer:<br \/>\nx = 10<\/p>\n<p>Explanation:<br \/>\nDE = \\(\\frac { 1 }{ 2 } \\)AB<br \/>\n5 = \\(\\frac { 1 }{ 2 } \\)x<br \/>\nx = 10<\/p>\n<p>Question 9.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43040\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-103.png\" alt=\"Big Ideas Math Geometry Solutions Chapter 6 Relationships Within Triangles 103\" width=\"173\" height=\"91\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49028\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-9.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 9\" width=\"110\" height=\"64\" \/><\/p>\n<p>Question 10.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43039\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-104.png\" alt=\"Big Ideas Math Geometry Solutions Chapter 6 Relationships Within Triangles 104\" width=\"180\" height=\"89\" \/><\/p>\n<p>Answer:<br \/>\nx = 8<\/p>\n<p>Explanation:<br \/>\nBE = EC<br \/>\nx = 8<\/p>\n<p>In Exercise 11-16, \\(\\overline{X J} \\cong \\overline{J Y}\\), \\(\\overline{Y L} \\cong \\overline{L Z}\\), and \\(\\overline{X K} \\cong \\overline{K Z}\\). Copy and complete the statement.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-43038\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-105.png\" alt=\"Big Ideas Math Geometry Solutions Chapter 6 Relationships Within Triangles 105\" width=\"270\" height=\"158\" \/><\/p>\n<p>Question 11.<br \/>\n\\(\\overline{J K}\\) || __________<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49029\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-11.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 11\" width=\"86\" height=\"39\" \/><\/p>\n<p>Question 12.<br \/>\n\\(\\overline{X Y}\\) || __________<\/p>\n<p>Answer:<br \/>\n\\(\\overline{X Y}\\) || KL<\/p>\n<p>Question 13.<br \/>\n\\(\\overline{J L}\\) || __________<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49030\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-13.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 13\" width=\"88\" height=\"39\" \/><\/p>\n<p>Question 14.<br \/>\n\\(\\overline{J L}\\) \u2245 __________ \u2245 __________<\/p>\n<p>Answer:<br \/>\n\\(\\overline{J L}\\) \u2245 JK \u2245 KL<\/p>\n<p>Question 15.<br \/>\n\\(\\overline{J Y}\\) \u2245 __________ \u2245 __________<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49031\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-15.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 15\" width=\"157\" height=\"35\" \/><\/p>\n<p>Question 16.<br \/>\n\\(\\overline{J K}\\) \u2245 __________ \u2245 __________<\/p>\n<p>Answer:<br \/>\n\\(\\overline{J K}\\) \u2245 JL \u2245 KL<\/p>\n<p><strong>MATHEMATICAL CONNECTIONS<\/strong><br \/>\nIn Exercises 17-19. use \u2206GHJ, where A, B, and C are midpoints of the sides.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43037\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-106.png\" alt=\"Big Ideas Math Geometry Solutions Chapter 6 Relationships Within Triangles 106\" width=\"143\" height=\"122\" \/><\/p>\n<p>Question 17.<br \/>\nWhen AB = 3x + 8 and GJ = 2x + 24, what is AB?<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49032\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-17.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 17\" width=\"226\" height=\"324\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-17.png 226w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-17-209x300.png 209w\" sizes=\"(max-width: 226px) 100vw, 226px\" \/><\/p>\n<p>Question 18.<br \/>\nWhen AC = 3y &#8211; 5 and HJ = 4y + 2, what is HB?<\/p>\n<p>Answer:<br \/>\nHB = 13<\/p>\n<p>Explanation:<br \/>\nAC = 0.5(HJ)<br \/>\n3y &#8211; 5 = 0.5(4y + 2)<br \/>\n3y &#8211; 5 = 2y + 1<br \/>\n3y &#8211; 2y = 1 + 5<br \/>\ny = 6<br \/>\nHB = 0.5(HJ)<br \/>\n= 0.5(4(6) + 2) = 0.5(26) = 13<\/p>\n<p>Question 19.<br \/>\nWhen GH = 7 &#8211; 1 and CB = 4z &#8211; 3. what is GA?<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49033\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-19.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 19\" width=\"409\" height=\"324\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-19.png 409w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-19-300x238.png 300w\" sizes=\"(max-width: 409px) 100vw, 409px\" \/><\/p>\n<p>Question 20.<br \/>\n<strong>ERROR ANALYSIS<\/strong><br \/>\nDescribe and correct the error.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43036\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-107.png\" alt=\"Big Ideas Math Geometry Solutions Chapter 6 Relationships Within Triangles 107\" width=\"344\" height=\"104\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-107.png 344w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-107-300x91.png 300w\" sizes=\"(max-width: 344px) 100vw, 344px\" \/><\/p>\n<p>Answer:<br \/>\nBC = \\(\\frac { 1 }{ 2 } \\)DE<br \/>\n10 = \\(\\frac { 12 }{ 2 } \\)(5)<br \/>\nAccording to triangle midsegment theorem, AD = DB and AE = EC.<\/p>\n<p>Question 21.<br \/>\n<strong>MODELING WITH MATHEMATICS<\/strong><br \/>\nThe distance between consecutive bases on a baseball held is 90 feet. A second baseman stands halfway between first base and second base, a shortstop stands hallway between second base and third base, and a pitcher stands halfway between first base and third base. Find the distance between the shortstop and the pitcher.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43035\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-108.png\" alt=\"Big Ideas Math Geometry Solutions Chapter 6 Relationships Within Triangles 108\" width=\"247\" height=\"225\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49034\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-21.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 21\" width=\"580\" height=\"168\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-21.png 580w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-21-300x87.png 300w\" sizes=\"(max-width: 580px) 100vw, 580px\" \/><\/p>\n<p>Question 22.<br \/>\n<strong>PROVING A THEOREM<\/strong><br \/>\nUse the figure from Example 2 to prose the Triangle Midsegment Theorem (Theorem 6.8) for midsegment \\(\\overline{D F}\\), where F is the midpoint of \\(\\overline{O C}\\).<br \/>\nAnswer:<\/p>\n<p>Question 23.<br \/>\n<strong>CRITICAL THINKING<\/strong><br \/>\n\\(\\overline{X Y}\\) is a midsegment of \u2206LMN. Suppose \\(\\overline{D E}\\) is called a \u201cquarter segment\u201d of \u2206LMN. What do you think an \u201ceighth segment\u201d would be? Make conjectures about the properties of a quarter segment and an eighth segment. Use variable coordinates to verify your conjectures.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43034\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-109.png\" alt=\"Big Ideas Math Geometry Solutions Chapter 6 Relationships Within Triangles 109\" width=\"239\" height=\"144\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49035\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-23.1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 23.1\" width=\"576\" height=\"253\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-23.1.png 576w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-23.1-300x132.png 300w\" sizes=\"(max-width: 576px) 100vw, 576px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49036\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-23.2.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 23.2\" width=\"615\" height=\"611\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-23.2.png 615w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-23.2-300x298.png 300w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-23.2-150x150.png 150w\" sizes=\"(max-width: 615px) 100vw, 615px\" \/><\/p>\n<p>Question 24.<br \/>\n<strong>THOUGHT PROVOKING<\/strong><br \/>\nFind a real-life object that uses midsegments as part of its structure. Print a photograph of the object and identify the midsegments of one of the triangles in the structure.<br \/>\nAnswer:<\/p>\n<p>Question 25.<br \/>\n<strong>ABSTRACT REASONING<\/strong><br \/>\nTo create the design shown. shade the triangle formed by the three midsegments of the triangle. Then repeat the process for each unshaded triangle.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43033\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-110.png\" alt=\"Big Ideas Math Geometry Solutions Chapter 6 Relationships Within Triangles 110\" width=\"251\" height=\"253\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-110.png 251w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-110-150x150.png 150w\" sizes=\"(max-width: 251px) 100vw, 251px\" \/><br \/>\na. What is the perimeter of the shaded triangle in Stage 1?<br \/>\nb. What is the total perimeter of all the shaded triangles in Stage 2?<br \/>\nc. What is the total perimeter of all the shaded triangles in Stage 3?<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49037\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-25.1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 25.1\" width=\"392\" height=\"319\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-25.1.png 392w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-25.1-300x244.png 300w\" sizes=\"(max-width: 392px) 100vw, 392px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49038\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-25.2.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 25.2\" width=\"530\" height=\"636\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-25.2.png 530w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-25.2-250x300.png 250w\" sizes=\"(max-width: 530px) 100vw, 530px\" \/><\/p>\n<p>Question 26.<br \/>\n<strong>HOW DO YOU SEE IT?<\/strong><br \/>\nExplain how you know that the yellow triangle is the midsegment triangle of the red triangle in the pattern of floor tiles shown.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43032\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-6-Relationships-Within-Triangles-111.png\" alt=\"Big Ideas Math Geometry Solutions Chapter 6 Relationships Within Triangles 111\" width=\"248\" height=\"128\" \/><\/p>\n<p>Answer:<br \/>\nBecause the yellow triangle vertices are located at the midpoints of the red triangle sides.<\/p>\n<p>Question 27.<br \/>\n<strong>ATTENDING TO PRECISION<\/strong><br \/>\nThe points P(2, 1), Q(4, 5), and R(7, 4) are the midpoints of the sides of a triangle. Graph the three midsegments. Then show how to use your graph and the properties of midsemeriLs to draw the original triangle. Give the coordinates of each vertex.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49039\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-27.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 27\" width=\"597\" height=\"484\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-27.png 597w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-27-300x243.png 300w\" sizes=\"(max-width: 597px) 100vw, 597px\" \/><\/p>\n<p>Maintaining Mathematical Proficiency<\/p>\n<p>Find a counter example to show that the conjecture is false.<\/p>\n<p>Question 28.<br \/>\nThe difference of two numbers is always less than the greater number.<\/p>\n<p>Answer:<br \/>\nWhen we subtract -3 from 4 i.e 4 -(-3) = 7. The number is greater than the individual integers. So, the subtraction of a negative number from a positive number gives an answer greater than both numbers.<\/p>\n<p>Question 29.<br \/>\nAn isosceles triangle is always equilateral.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49040\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-29.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.4 Question 29\" width=\"569\" height=\"90\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-29.png 569w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.4-Question-29-300x47.png 300w\" sizes=\"(max-width: 569px) 100vw, 569px\" \/><\/p>\n<h3><a id=\"6.5_Indirect_Proof_and_Inequalities_in_One_Triangle\"><\/a>6.5 Indirect Proof and Inequalities in One Triangle<\/h3>\n<p><strong>Exploration 1<\/strong><\/p>\n<p>Comparing Angle Measures and Side Lengths<\/p>\n<p>Work with a partner: Use dynamic geometry software. Draw any scalene \u2206ABC<\/p>\n<p>a. Find the side lengths and angle measures of the triangle.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43158\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-112.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 112\" width=\"517\" height=\"242\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-112.png 517w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-112-300x140.png 300w\" sizes=\"(max-width: 517px) 100vw, 517px\" \/><br \/>\nAnswer:<\/p>\n<p>b. Order the side lengths. Order the angle measures. What do you observe?<br \/>\nAnswer:<\/p>\n<p>c. Drug the vertices of \u2206ABC to form new triangles. Record the side lengths and angle measures in a table. Write a conjecture about your findings.<br \/>\nAnswer:<\/p>\n<p><strong>Exploration 2<\/strong><\/p>\n<p>A Relationship of the Side Lengths of a Triangle<\/p>\n<p>Work with a partner. Use dynamic geometry software. Draw any \u2206ABC.<\/p>\n<p>a. Find the side lengths of the triangle.<br \/>\nAnswer:<\/p>\n<p>b. Compare each side length with the sum of the other two side lengths.<br \/>\n<strong>ATTENDING TO PRECISION<\/strong><br \/>\nTo be proficient in math, you need to express numerical answers with a degree of precision appropriate for the content.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43157\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-113.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 113\" width=\"418\" height=\"246\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-113.png 418w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-113-300x177.png 300w\" sizes=\"(max-width: 418px) 100vw, 418px\" \/><br \/>\nAnswer:<\/p>\n<p>c. Drag the vertices of \u2206ABC to form new triangles and repeat parts (a) and (b). Organize your results in a table. Write a conjecture about your findings.<br \/>\nAnswer:<\/p>\n<p>Communicate Your Answer<\/p>\n<p>Question 3.<br \/>\nHow are the sides related to the angles of a triangle? How are any two sides of a triangle related to the third side?<br \/>\nAnswer:<\/p>\n<p>Question 4.<br \/>\nIs it possible for a triangle to have side lengths of 3, 4, and 10? Explain.<br \/>\nAnswer:<\/p>\n<h3><a id=\"Lesson_6.5_Indirect_Proof_and_Inequalities_in_One_Triangle\"><\/a>Lesson 6.5 Indirect Proof and Inequalities in One Triangle<\/h3>\n<p><strong>Monitoring Progress<\/strong><\/p>\n<p>Question 1.<br \/>\nWrite in indirect proof that a scalene triangle cannot have two congruent angles.<\/p>\n<p>Answer:<\/p>\n<p>Question 2.<br \/>\nList the angle of \u2206PQR in order from smallest to largest.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43156\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-114.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 114\" width=\"121\" height=\"121\" \/><\/p>\n<p>Answer:<br \/>\nThe sides from the largest to smallest are PQ, QR, PR<br \/>\nThe angles opposite to sides are\u00a0\u2220R, \u2220P, \u2220Q<br \/>\nThe angles from smallest to the largest are \u2220Q, \u2220P, \u2220R<\/p>\n<p>Question 3.<br \/>\nList the sides of \u2206RST in order from shortest to longest.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43155\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-115.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 115\" width=\"231\" height=\"96\" \/><\/p>\n<p>Answer:<br \/>\nThe angles from shortest to longest are \u2220R, \u2220T, \u2220S<\/p>\n<p>Question 4.<br \/>\nA triangle has one side of length 12 inches and another side of length 20 inches. Describe the possible lengths of the third side.<\/p>\n<p>Answer:<br \/>\nThe length of the third side must be greater than 8 and lesser than 32.<\/p>\n<p>Explanation:<br \/>\nLet x represent the length of the third side. Draw diagrams to help visualize the small and large values of x. Then use the Triangle Inequality Theorem to write and solve inequalities.<br \/>\nThe smallest value of x is x + 12 &gt; 20. So, x &gt; 8<br \/>\nThe largest value of x is 12 + 20 &gt; x<br \/>\n32 &gt; x<br \/>\nSo, 32 &gt; x or 32 &lt; x<br \/>\nThe length of the third side must be greater than 8 and lesser than 32.<\/p>\n<p>Decide Whether it is possible to construct a triangle with the given side lengths. Explain your reasoning.<\/p>\n<p>Question 5.<br \/>\n4 ft, 9 ft, 10 ft<\/p>\n<p>Answer:<br \/>\nIt is not possible to construct a triangle.<\/p>\n<p>Explanation:<br \/>\nAccording to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle is greater than the length of the third side.<br \/>\nSo, 9 + 4 = 13<br \/>\n13 is not greater than 10.<br \/>\nSo, it is not possible to construct a triangle.<\/p>\n<p>Question 6.<br \/>\n8 ft, 9 ft, 18 ft<\/p>\n<p>Answer:<br \/>\nIt is possible to construct a triangle.<\/p>\n<p>Explanation:<br \/>\nAccording to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle is greater than the length of the third side.<br \/>\n8 + 9 = 17 &gt; 18<br \/>\nSo, it is possible to construct a triangle.<\/p>\n<p>Question 7.<br \/>\n5 cm, 7 cm, 12 cm<\/p>\n<p>Answer:<br \/>\nIt is not possible to construct a triangle.<\/p>\n<p>Explanation:<br \/>\nAccording to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle is greater than the length of the third side.<br \/>\n7 + 5 = 12<br \/>\n12 is not greater than 12<br \/>\nso, it is not possible to construct a triangle.<\/p>\n<h3><a id=\"Exercise_6.5_Indirect_Proof_and_Inequalities_in_One_Triangle\"><\/a>Exercise 6.5 Indirect Proof and Inequalities in One Triangle<\/h3>\n<p>Vocabulary and Core Concept Check<\/p>\n<p>Question 1.<br \/>\n<strong>VOCABULARY<\/strong><br \/>\nWhy is an indirect proof also called a proof by contradiction?<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49041\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 1\" width=\"559\" height=\"87\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-1.png 559w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-1-300x47.png 300w\" sizes=\"(max-width: 559px) 100vw, 559px\" \/><\/p>\n<p>Question 2.<br \/>\n<strong>WRITING<\/strong><br \/>\nHow can you tell which side of a triangle is the longest Irom the angle measures of the<br \/>\ntriangle? How can you tell which side is the shortest?<\/p>\n<p>Answer:<br \/>\nThe angle which is opposite to the longest side is the longest angle.<\/p>\n<p>Monitoring progress and Modeling with Mathematics<\/p>\n<p>In Exercises 3-6, write the first step in an indirect proof of the statement.<\/p>\n<p>Question 3.<br \/>\nIf WV + VU \u2260 12 inches and VU = 5 inches, then WV \u2260 7 inches.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49042\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-3.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 3\" width=\"385\" height=\"36\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-3.png 385w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-3-300x28.png 300w\" sizes=\"(max-width: 385px) 100vw, 385px\" \/><\/p>\n<p>Question 4.<br \/>\nIf x and y are odd integers. then xy is odd.<\/p>\n<p>Answer:<br \/>\nAssume x = 1, y = 3<br \/>\nxy = 3<\/p>\n<p>Question 5.<br \/>\nIn \u2206ABC. if m\u2220A = 100\u00b0, then \u2220B is not a right angle.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49043\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-5.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 5\" width=\"420\" height=\"32\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-5.png 420w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-5-300x23.png 300w\" sizes=\"(max-width: 420px) 100vw, 420px\" \/><\/p>\n<p>Question 6.<br \/>\nIn \u2206JKL, if M is the midpoint of \\(\\overline{K L}\\), then \\(\\overline{J M}\\) is a median.<\/p>\n<p>Answer:<br \/>\nAssume temporarily JM is a median.<\/p>\n<p>In Exercises 7 and 8, determine which two statements contradict each other. Explain your reasoning.<\/p>\n<p>Question 7.<br \/>\n(A) \u2206LMN is a right triangle.<br \/>\n(B) \u2220L \u2245\u2220V<br \/>\n(C) \u2206LMN is equilateral.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49044\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-7.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 7\" width=\"573\" height=\"59\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-7.png 573w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-7-300x31.png 300w\" sizes=\"(max-width: 573px) 100vw, 573px\" \/><\/p>\n<p>Question 8.<br \/>\n(A) Both \u2220X and \u2220Y have measures greater than 20\u00b0.<br \/>\n(B) Both \u2220X and \u2220Y have measures less than 30\u00b0.<br \/>\n(C) m\u2220X + m\u2220Y = 62\u00b0<br \/>\nAnswer:<br \/>\nB and C. If \u2220X and \u2220Y are less than 30, then\u00a0 m\u2220X + m\u2220Y is not equal to 62.<\/p>\n<p>In Exercises 9 and 10, use a ruler and protractor to draw the given type of triangle. Mark the largest angle and longest side in red and the smallest angle and shortest side in blue. What do you notice?<\/p>\n<p>Question 9.<br \/>\nacute scalene<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49045\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-9.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 9\" width=\"574\" height=\"363\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-9.png 574w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-9-300x190.png 300w\" sizes=\"(max-width: 574px) 100vw, 574px\" \/><\/p>\n<p>Question 10.<br \/>\nright scalene<\/p>\n<p>Answer:<br \/>\nTo construct a scalene triangle draw a segment and label it AB. Assuring that three sides have different lengths. Draw an arc with center A and AC as radius draw an arc and draw another arc taking BC as radius and B as centre. Label the point of intersection of two arcs as C.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-107005\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-answers-geometry-chapter-6-relationships-within-triangles-6.5-1.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 6.5 1\" width=\"255\" height=\"328\" \/><br \/>\nThe largest side is BC. So, the largest angle is \u2220A. The smallest side is AB and the smallest angle is \u2220C.<\/p>\n<p>In Exercises 11 and 12, list the angles of the given triangle from smallest to largest.<\/p>\n<p>Question 11.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43154\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-116.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 116\" width=\"145\" height=\"99\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49046\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-11.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 11\" width=\"550\" height=\"89\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-11.png 550w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-11-300x49.png 300w\" sizes=\"(max-width: 550px) 100vw, 550px\" \/><\/p>\n<p>Question 12.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43153\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-117.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 117\" width=\"145\" height=\"96\" \/><br \/>\nAnswer:<br \/>\nThe sides of \u25b3JKL from smallest to largest are KL, JL, JK. So by the triangle, longer side theorem, the angles from smallest to largest are \u2220J, \u2220K, \u2220L.<\/p>\n<p>In Exercises 13-16, list the sides of the given triangle from shortest to longest.<\/p>\n<p>Question 13.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43152\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-118.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 118\" width=\"144\" height=\"117\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49047\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-13.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 13\" width=\"567\" height=\"89\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-13.png 567w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-13-300x47.png 300w\" sizes=\"(max-width: 567px) 100vw, 567px\" \/><\/p>\n<p>Question 14.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43151\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-119.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 119\" width=\"155\" height=\"119\" \/><br \/>\nAnswer:<br \/>\nThe angles of the triangle from smallest to largest are \u2220Z, \u2220X, \u2220Y. So by the trinagle larger angle theorem, the sides from shortest to largest are XY, YZ, and XZ.<\/p>\n<p>Question 15.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43150\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-120.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 120\" width=\"181\" height=\"107\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49048\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-15.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 15\" width=\"575\" height=\"123\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-15.png 575w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-15-300x64.png 300w\" sizes=\"(max-width: 575px) 100vw, 575px\" \/><\/p>\n<p>Question 16.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43149\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-121.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 121\" width=\"149\" height=\"122\" \/><br \/>\nAnswer:<br \/>\n\u2220D = 180 &#8211; (90 + 33) = 180 &#8211; 57<br \/>\nThe angles from smallest to largest are \u2220G, \u2220D, \u2220F. So by the triangle larger angle theorem, the sides from shortest to longest are FD, FG, and DG.<\/p>\n<p>In Exercises 17-20, describe the possible lengths of the third side of the triangle given the lengths of the other to sides.<\/p>\n<p>Question 17.<br \/>\n5 inches, 12 inches<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49049\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-17.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 17\" width=\"499\" height=\"196\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-17.png 499w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-17-300x118.png 300w\" sizes=\"(max-width: 499px) 100vw, 499px\" \/><\/p>\n<p>Question 18.<br \/>\n12 feet, 18 feet<\/p>\n<p>Answer:<br \/>\nThe possible lengths of the third side are greater than 6 and less than 30 ft<\/p>\n<p>Explanation:<br \/>\nx + 12 &gt; 18<br \/>\nx &gt; 6<br \/>\n12 + 18 &gt; x<br \/>\n30 &gt; x or x &lt; 30<br \/>\nThe possible lengths of the third side are greater than 6 and less than 30 ft<\/p>\n<p>Question 19.<br \/>\n2 feet, 40 inches<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49050\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-19.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 19\" width=\"493\" height=\"202\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-19.png 493w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-19-300x123.png 300w\" sizes=\"(max-width: 493px) 100vw, 493px\" \/><\/p>\n<p>Question 20.<br \/>\n25 meters, 25 meters<\/p>\n<p>Answer:<br \/>\nThe possible lengths of the third side are greater than 0 and less than 50 m.<\/p>\n<p>Explanation:<br \/>\nx + 25 &gt; 25<br \/>\nx &gt; 0<br \/>\n25 + 25 &gt; x<br \/>\n50 &gt; x or x &lt; 50<br \/>\nThe possible lengths of the third side are greater than 0 and less than 50 m.<\/p>\n<p>In Exercises 21-23, is it possible to construct a triangle with the given side lengths? If not, explain why not.<\/p>\n<p>Question 21.<br \/>\n6, 7, 11<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49051\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-21.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 21\" width=\"548\" height=\"164\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-21.png 548w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-21-300x90.png 300w\" sizes=\"(max-width: 548px) 100vw, 548px\" \/><\/p>\n<p>Question 22.<br \/>\n3, 6, 9<\/p>\n<p>Answer:<br \/>\nNo<\/p>\n<p>Explanation:<br \/>\n3 + 6 = 9 \u2192 9 = 9<br \/>\n6 + 9 = 15 \u2192 15 &lt; 9<br \/>\n9 + 3 = 12 \u2192 12 &lt; 9<br \/>\nNo, the sum of any two side lengths of a triangle is less than the length of the third side.<\/p>\n<p>Question 23.<br \/>\n28, 17, 46<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49052\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-23.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 23\" width=\"459\" height=\"34\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-23.png 459w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-23-300x22.png 300w\" sizes=\"(max-width: 459px) 100vw, 459px\" \/><\/p>\n<p>Question 24.<br \/>\n35, 120, 125<\/p>\n<p>Answer:<br \/>\n35 + 120 = 155 \u2192 155 &lt; 125<br \/>\nSo, No.<\/p>\n<p>Question 25.<br \/>\n<strong>ERROR ANALYSIS<\/strong><br \/>\nDescribe and correct the error in writing the first step of an indirect proof.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43148\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-122.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 122\" width=\"327\" height=\"95\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-122.png 327w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-122-300x87.png 300w\" sizes=\"(max-width: 327px) 100vw, 327px\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49053\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-25.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 25\" width=\"540\" height=\"61\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-25.png 540w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-25-300x34.png 300w\" sizes=\"(max-width: 540px) 100vw, 540px\" \/><\/p>\n<p>Question 26.<br \/>\n<strong>ERROR ANALYSIS<\/strong><br \/>\nDescribe and correct the error in labeling the side lengths 1, 2, and \u221a3 on the triangle.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43147\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-123.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 123\" width=\"268\" height=\"131\" \/><\/p>\n<p>Answer:<br \/>\nNumber 1 is opposite to 90 degrees and 2 is opposite to 60 degrees and the remaining is \u221a3 on the triangle.<\/p>\n<p>Question 27.<br \/>\n<strong>REASONING<\/strong><br \/>\nYou are a lawyer representing a client who has been accused of a crime. The crime took place in Los Angeles, California. Security footage shows your client in New York at the time of the crime. Explain how to use indirect reasoning to prove your client is innocent.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49054\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-27.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 27\" width=\"555\" height=\"141\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-27.png 555w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-27-300x76.png 300w\" sizes=\"(max-width: 555px) 100vw, 555px\" \/><\/p>\n<p>Question 28.<br \/>\n<strong>REASONING<\/strong><br \/>\nYour class has fewer than 30 students. The teacher divides your class into two groups. The first group has 15 students. Use indirect reasoning to show that the second group must have fewer than 15 students.<\/p>\n<p>Answer:<br \/>\nThe assumption that N2 \u2265 15 is false.<\/p>\n<p>Explanation:<br \/>\nThe number of students in a class N &lt; 30<br \/>\nThe number of students in the first group N1 = 15<br \/>\nThe number of students in the second group N2<br \/>\nN = N1 + N2<br \/>\nAssume temporarily that the number of student in the second group has 15 or more students<br \/>\nN2 \u2265 15<br \/>\nN1 + N2 \u2265 15 + 15<br \/>\n30 \u2265 30<br \/>\nSo, the assumption that N2 \u2265 15 is false.<\/p>\n<p>Question 29.<br \/>\n<strong>PROBLEM SOLVING<\/strong><br \/>\nWhich statement about \u2206TUV is false?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43146\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-124.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 124\" width=\"128\" height=\"137\" \/><br \/>\n(A) UV &gt; TU<br \/>\n(B) UV + TV &gt; TU<br \/>\n(C) UV &lt; TV<br \/>\n(D) \u2206TUV is isosceles.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49055\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-29.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 29\" width=\"580\" height=\"87\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-29.png 580w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-29-300x45.png 300w\" sizes=\"(max-width: 580px) 100vw, 580px\" \/><\/p>\n<p>Question 30.<br \/>\n<strong>PROBLEM SOLVING<\/strong><br \/>\nIn \u2206RST. which is a possible side length for ST? Select all that apply.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43145\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-125.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 125\" width=\"115\" height=\"117\" \/><br \/>\n(A) 7<br \/>\n(B) 8<br \/>\n(C) 9<br \/>\n(D) 10<br \/>\nAnswer:<br \/>\nA.<\/p>\n<p>Explanation:<br \/>\n\u2220R = 180 &#8211; (65 + 56) = 180 &#8211; 121 = 59, which indicates that \u2206RST i isosceles. By the trinagle longer side theorem, ST is 7<\/p>\n<p>Question 31.<br \/>\n<strong>PROOF<\/strong><br \/>\nWrite an indirect proof that an odd number is not divisible by 4.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49056\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-31.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 31\" width=\"554\" height=\"265\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-31.png 554w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-31-300x144.png 300w\" sizes=\"(max-width: 554px) 100vw, 554px\" \/><\/p>\n<p>Question 32.<br \/>\n<strong>PROOF<\/strong><br \/>\nWrite an indirect proof of the statement<br \/>\n&#8220;In \u2206QRS, if m\u2220Q + m\u2220R = 90\u00b0, then m\u2220S = 90\u00b0.&#8221;<br \/>\nAnswer:<\/p>\n<p>Question 33.<br \/>\n<strong>WRITING<\/strong><br \/>\nExplain why the hypotenuse of a right triangle must always be longer than either leg.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49057\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-33.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 33\" width=\"603\" height=\"214\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-33.png 603w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-33-300x106.png 300w\" sizes=\"(max-width: 603px) 100vw, 603px\" \/><\/p>\n<p>Question 34.<br \/>\n<strong>CRITICAL THINKING<\/strong><br \/>\nIs it possible to decide if three side lengths form a triangle without checking all three inequalities shown in the Triangle Inequality Theorem (Theorem 6. 11)? Explain your reasoning.<br \/>\nAnswer:<\/p>\n<p>Question 35.<br \/>\nM<strong>ODELING WITH MATHEMATICS<\/strong><br \/>\nYou can estimate the width of the river from point A to the tree at point B by measuring the angle to the tree at several locations along the riverbank. The diagram shows the results for locations C and D.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43144\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-126.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 126\" width=\"322\" height=\"170\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-126.png 322w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-126-300x158.png 300w\" sizes=\"(max-width: 322px) 100vw, 322px\" \/><br \/>\na. Using \u2206BCA and \u2206BDA, determine the possible widths of the river. Explain your reasoning.<br \/>\nb. What could you do if you wanted a closer estimate?<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49058\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-35.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 35\" width=\"581\" height=\"341\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-35.png 581w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-35-300x176.png 300w\" sizes=\"(max-width: 581px) 100vw, 581px\" \/><\/p>\n<p>Question 36.<br \/>\n<strong>MODELING WITH MATHEMATICS<\/strong><br \/>\nYou travel from Fort Peck Lake to Glacier National Park and from Glacier National Park to Granite Peak.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43143\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-127.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 127\" width=\"337\" height=\"196\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-127.png 337w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-127-300x174.png 300w\" sizes=\"(max-width: 337px) 100vw, 337px\" \/><br \/>\na. Write two inequalities to represent the possible distances from Granite Peak back to Fort Peck Lake.<br \/>\nAnswer:<\/p>\n<p>b. How is your answer to part (a) affected if you know that m\u22202 &lt; m\u22201 and m\u22202 &lt; m\u22203?<br \/>\nAnswer:<\/p>\n<p>Question 37.<br \/>\n<strong>REASONING<\/strong><br \/>\nIn the figure. \\(\\overline{X Y}\\) bisects \u2220WYZ. List all six angles of \u2206XYZ and \u2206WXY in order from smallest to largest. Explain \u2018our reasoning.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43142\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-128.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 128\" width=\"112\" height=\"95\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49059\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-37.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 37\" width=\"589\" height=\"256\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-37.png 589w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-37-300x130.png 300w\" sizes=\"(max-width: 589px) 100vw, 589px\" \/><\/p>\n<p>Question 38.<br \/>\n<strong>MATHEMATICAL CONNECTIONS<\/strong><br \/>\nIn \u2206DEF, m\u2220D = (x + 25)\u00b0. m\u2220E = (2x &#8211; 4)\u00b0, and in m\u2220F = 63\u00b0. List the side lengths and angle measures of the triangle in order from least to greatest.<\/p>\n<p>Answer:<\/p>\n<p>Explanation:<br \/>\nsum of angles is 180 degrees<br \/>\nx + 25 + 2x &#8211; 4 + 63 = 180<br \/>\n3x + 84 = 180<br \/>\n3x = 96<br \/>\nx = 32<br \/>\nm\u2220D = 32 + 25 = 57<br \/>\nm\u2220E = 2(32) &#8211; 4 = 60<br \/>\nm\u2220F = 63<br \/>\nThwe angles from least to greatest are m\u2220D, m\u2220E, m\u2220F and the side lengths are EF, DF, DE.<\/p>\n<p>Question 39.<br \/>\n<strong>ANALYZING RELATIONSHIPS<\/strong><br \/>\nAnother triangle inequality relationship is given by the Exterior Angle Inequality Theorem. It states:<\/p>\n<p>The measures of an exterior angle of a triangle is greater than the measure of either of the nonadjacent interior angles.<\/p>\n<p>Explain how you know that m\u22201 &gt; m\u2220A and m\u22201 &gt; m\u2220B in \u2206ABC with exterior angle \u22201.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43141\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-129.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 129\" width=\"109\" height=\"85\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49060\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-39.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 39\" width=\"580\" height=\"306\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-39.png 580w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-39-300x158.png 300w\" sizes=\"(max-width: 580px) 100vw, 580px\" \/><\/p>\n<p><strong>MATHEMATICAL CONNECTIONS<\/strong><br \/>\nIn Exercises 40 and 41, describe the possible values of x.<\/p>\n<p>Question 40.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43140\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-130.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 130\" width=\"142\" height=\"98\" \/><br \/>\nAnswer:<br \/>\nThe possible values of x are x &gt; \\(\\frac { 5 }{ 3 } \\) and x &lt; 15<\/p>\n<p>Explanation:<br \/>\nJK + JL &gt; KL<br \/>\nx + 11 + 5x &#8211; 9 &gt; 2x +10<br \/>\n6x + 2 &gt; 2x + 10<br \/>\n4x &gt; 8<br \/>\nx &gt; 2<br \/>\nJK + KL &gt; JL<br \/>\nx + 11 + 2x + 10 &gt; 5x &#8211; 9<br \/>\n3x + 21 &gt; 5x &#8211; 9<br \/>\n30 &gt; 2x<br \/>\n15 &gt; x<br \/>\nJL + KL &gt; JK<br \/>\n5x &#8211; 9 + 2x + 10 &gt; x + 11<br \/>\n7x + 1 &gt; x + 11<br \/>\n6x &gt; 10<br \/>\nx &gt; \\(\\frac { 5 }{ 3 } \\)<br \/>\nThe possible values of x are x &gt; \\(\\frac { 5 }{ 3 } \\) and x &lt; 15<\/p>\n<p>Question 41.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43139\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-131.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 131\" width=\"140\" height=\"112\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49061\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-41.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 41\" width=\"446\" height=\"584\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-41.png 446w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-41-229x300.png 229w\" sizes=\"(max-width: 446px) 100vw, 446px\" \/><\/p>\n<p>Question 42.<br \/>\n<strong>HOW DO YOU SEE IT?<\/strong><br \/>\nYour house is on the corner of Hill Street and Eighth Street. The library is on the corner of View Street and Seventh Street. What is the shortest route to et from your house to the library? Explain your reasoning.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43138\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-132.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 132\" width=\"250\" height=\"157\" \/><br \/>\nAnswer:<br \/>\nThrough Washington ave is the shortest route to reach from house to the library.<\/p>\n<p>Question 43.<br \/>\n<strong>PROVING A THEOREM<\/strong><br \/>\nUse the diagram to prove the Triangle Longer Side Theorem (Theorem 6.9).<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43137\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-133.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 133\" width=\"184\" height=\"81\" \/><br \/>\nGiven BC &gt; AB, BD = BA<br \/>\nProve m\u2220BAC &gt; m\u2220C<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49062\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-43.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 43\" width=\"578\" height=\"410\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-43.png 578w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-43-300x213.png 300w\" sizes=\"(max-width: 578px) 100vw, 578px\" \/><\/p>\n<p>Question 44.<br \/>\n<strong>USING STRUCTURE<\/strong><br \/>\nThe length of the base of an isosceles triangle is l. Describe the possible lengths for each leg. Explain our reasoning.<br \/>\nAnswer:<br \/>\nIf other lengths are x and x<br \/>\nThen, x + x &gt; l<br \/>\n2x &gt; l<br \/>\nx &gt; l\/2<br \/>\nx + l &gt; x<br \/>\nThe possible lengths of each leg are l are l\/2.<\/p>\n<p>Question 45.<br \/>\n<strong>MAKING AN ARGUMENT<\/strong><br \/>\nYour classmate claims to have drawn a triangle with one side length of 13 inches and a perimeter of 2 feet. Is this possible? Explain your reasoning.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49063\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-45.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 45\" width=\"550\" height=\"110\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-45.png 550w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-45-300x60.png 300w\" sizes=\"(max-width: 550px) 100vw, 550px\" \/><\/p>\n<p>Question 46.<br \/>\n<strong>THOUGHT PROVOKING<\/strong><br \/>\nCut two pieces of string that are each 24 centimeters long. Construct an isosceles triangle out of one string and a scalene triangle out of the other. Measure and record the side lengths. Then classify each triangle by its angles.<br \/>\nAnswer:<\/p>\n<p>Question 47.<br \/>\n<strong>PROVING A THEOREM<\/strong><br \/>\nProve the Triangle Inequality Theorem (Theorem 6. 11).<br \/>\nGiven \u2206ABC<br \/>\nProve AB + BC &gt; AC, AC + BC &gt; AB, and AB + AC &gt; BC<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49064\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-47.1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 47.1\" width=\"521\" height=\"346\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-47.1.png 521w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-47.1-300x199.png 300w\" sizes=\"(max-width: 521px) 100vw, 521px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49065\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-47.2.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 47.2\" width=\"541\" height=\"496\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-47.2.png 541w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-47.2-300x275.png 300w\" sizes=\"(max-width: 541px) 100vw, 541px\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49066\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-47.3.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 47.3\" width=\"518\" height=\"209\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-47.3.png 518w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-47.3-300x121.png 300w\" sizes=\"(max-width: 518px) 100vw, 518px\" \/><\/p>\n<p>Question 48.<br \/>\n<strong>ATTENDING TO PRECISION<\/strong><br \/>\nThe perimeter of \u2206HGF must be between what two integers? Explain your reasoning.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43136\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-134.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 134\" width=\"176\" height=\"98\" \/><br \/>\nAnswer:<\/p>\n<p>Question 49.<br \/>\n<strong>PROOF<\/strong><br \/>\nWrite an indirect proof that a perpendicular segment is the shortest segment from a point to a plane.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43135\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-135.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 135\" width=\"207\" height=\"114\" \/><br \/>\nGiven \\(\\overline{P C}\\) \u22a5 palne M<br \/>\nProve \\(\\overline{P C}\\) is the shortest from P to plane M.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49067\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-49.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 49\" width=\"584\" height=\"256\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-49.png 584w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-49-300x132.png 300w\" sizes=\"(max-width: 584px) 100vw, 584px\" \/><\/p>\n<p>Maintaining Mathematical Proficiency<\/p>\n<p>Name the indicated angle between the pair of sides given.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-43134\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-6-Relationships-Within-Triangles-136.png\" alt=\"Big Ideas Math Answer Key Geometry Chapter 6 Relationships Within Triangles 136\" width=\"150\" height=\"90\" \/><\/p>\n<p>Question 50.<br \/>\n\\(\\overline{A E}\\) and \\(\\overline{B E}\\)<br \/>\nAnswer:<br \/>\nThe included angle is \\(\\overline{A E}\\) and \\(\\overline{B E}\\) is \u2220AEB<\/p>\n<p>Question 51.<br \/>\n\\(\\overline{A C}\\) and \\(\\overline{D C}\\)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49068\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-51.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 51\" width=\"475\" height=\"41\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-51.png 475w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-51-300x26.png 300w\" sizes=\"(max-width: 475px) 100vw, 475px\" \/><\/p>\n<p>Question 52.<br \/>\n\\(\\overline{A D}\\) and \\(\\overline{D C}\\)<br \/>\nAnswer:<br \/>\nThe included angle is \\(\\overline{A D}\\) and \\(\\overline{D C}\\) is \u2220ADC<\/p>\n<p>Question 53.<br \/>\n\\(\\overline{C E}\\) and \\(\\overline{B E}\\)<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49069\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-53.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.5 Question 53\" width=\"465\" height=\"36\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-53.png 465w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.5-Question-53-300x23.png 300w\" sizes=\"(max-width: 465px) 100vw, 465px\" \/><\/p>\n<h3><a id=\"6.6_Inequalities_in_Two_Triangles\"><\/a>6.6 Inequalities in Two Triangles<\/h3>\n<p><strong>Exploration 1<\/strong><\/p>\n<p>Comparing Measures in Triangles<\/p>\n<p>Work with a partner. Use dynamic geometry software.<\/p>\n<p>a. Draw \u2206ABC, as shown below.<br \/>\nAnswer:<\/p>\n<p>b. Draw the circle with center C(3, 3) through the point A(1, 3).<br \/>\nAnswer:<\/p>\n<p>c. Draw \u2206DBC so that D is a point on the circle.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43274\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-137.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 137\" width=\"440\" height=\"246\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-137.png 440w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-137-300x168.png 300w\" sizes=\"(max-width: 440px) 100vw, 440px\" \/><br \/>\nAnswer:<\/p>\n<p>d. Which two sides of \u2206ABC are congruent to two sides of \u2206DBC? Justify your answer.<br \/>\nAnswer:<\/p>\n<p>e. Compare the lengths of \\(\\overline{A B}\\) and \\(\\overline{D B}\\). Then compare the measures of \u2220ACB and \u2220DCB. Are the results what you expected? Explain.<br \/>\nAnswer:<\/p>\n<p>f. Drag point D to several locations on the circle. At each location, repeat part (e). Copy and record your results in the table below.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43273\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-138.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 138\" width=\"506\" height=\"208\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-138.png 506w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-138-300x123.png 300w\" sizes=\"(max-width: 506px) 100vw, 506px\" \/><br \/>\nAnswer:<\/p>\n<p>g. Look for a pattern of the measures in our table. Then write a conjecture that summarizes your observations.<br \/>\n<strong>CONSTRUCTING VIABLE ARGUMENTS<\/strong><br \/>\nTo be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures.<br \/>\nAnswer:<\/p>\n<p>Communicate Your Answer<\/p>\n<p>Question 2.<br \/>\nIf two sides of one triangle are congruent to two sides of another triangle, what can you say about the third sides of the triangles?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43272\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-139.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 139\" width=\"155\" height=\"177\" \/><br \/>\nAnswer:<\/p>\n<p>Question 3.<br \/>\nExplain how you can use the hinge shown at the left to model the concept described in Question 2.<br \/>\nAnswer:<\/p>\n<h3><a id=\"Lesson_6.6_Inequalities_in_Two_Triangles\"><\/a>Lesson 6.6 Inequalities in Two Triangles<\/h3>\n<p><strong>Monitoring Progress<\/strong><\/p>\n<p>Use the diagram<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-43271\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-140.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 140\" width=\"105\" height=\"110\" \/><\/p>\n<p>Question 1.<br \/>\nIf PR = PS and m\u2220QPR &gt; m\u2220QPS, which is longer, \\(\\overline{S Q}\\) or \\(\\overline{R Q}\\)?<\/p>\n<p>Answer:<br \/>\nGiven that,<br \/>\nPR = PS and PQ \u2245 PQ by the reflexive property of congruence.<br \/>\nm\u2220QPR &gt; m\u2220QPS.<br \/>\nSo, two sides of \u25b3PSQ are congruent to two sides of \u25b3PRQ and included angle in \u25b3PRQ is larger.<br \/>\nBy the hinge theorem RQ &gt; SQ.<\/p>\n<p>Question 2.<br \/>\nIf PR = PS and RQ &lt; SQ, which is larger, \u2220RPQ or \u2220SPQ?<\/p>\n<p>Answer:<br \/>\nGiven that,<br \/>\nPR = PS, RQ &lt; SQ<br \/>\nPQ \u2245 PQ by the reflexive property of congruence theorem.<br \/>\nSo, two sides of \u25b3PSQ are congruent to two sides of \u25b3PRQ and third side of \u25b3PSQ is longer.<br \/>\nBy the converse of the hinge theorem, \u2220SPQ &gt; \u2220RPQ.<\/p>\n<p>Question 3.<br \/>\nWrite a temporary assumption you can make to prove the Hinge Theorem indirectly. What two cases does that assumption lead to?<\/p>\n<p>Answer:<br \/>\nHinge theorem states that if two sides of the triangle are congruent to two sides of another triangle and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.<\/p>\n<p>Question 4.<br \/>\n<strong>WHAT IF?<\/strong><br \/>\nIn Example 5, Group C leaves camp and travels 2 miles due north. then turns 40\u00b0 towards east and travels 1.2 miles. Compare the distances from camp for all three groups.<\/p>\n<p>Answer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-107299\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-geometry-answers-chapter-6-relationships-within-triangles-6.6-1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.6 1\" width=\"362\" height=\"154\" \/><br \/>\ngroup A = 180 &#8211; 45 = 135<br \/>\ngroup B = 180 &#8211; 30 = 150<br \/>\nGroup C = 180 &#8211; 40 = 140<br \/>\nBecause 150 &gt; 140 &gt; 135, the distance Group B from the camp is greater than the distance Group C, A. by the hinge theorem.<br \/>\nThe distances are Group B &gt; group C &gt; group A.<\/p>\n<h3><a id=\"Exercise_6.6_Inequalities_in_Two_Triangles\"><\/a>Exercise 6.6 Inequalities in Two Triangles<\/h3>\n<p>Vocabulary and Core Concept Check<\/p>\n<p>Question 1.<br \/>\n<strong>WRITING<\/strong><br \/>\nExplain why Theorem 6.12 is named the \u201cHinge Theorem.\u201d<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49070\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-1.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.6 Question 1\" width=\"579\" height=\"142\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-1.png 579w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-1-300x74.png 300w\" sizes=\"(max-width: 579px) 100vw, 579px\" \/><\/p>\n<p>Question 2.<br \/>\n<strong>COMPLETE THE SENTENCE<\/strong><br \/>\nIn \u2206ABC and \u2206DEF, \\(\\overline{A B} \\cong \\overline{D E}\\), \\(\\overline{B C} \\cong \\overline{E F}\\). and AC &lt; DF. So m\u2220_______ &gt; m \u2220 _______ by the Converse of the Hinge Theorem<\/p>\n<p>Answer:<br \/>\nIn \u2206ABC and \u2206DEF, \\(\\overline{A B} \\cong \\overline{D E}\\), \\(\\overline{B C} \\cong \\overline{E F}\\). and AC &lt; DF. So m\u2220E &gt; m \u2220B by the Converse of the Hinge Theorem<\/p>\n<p>Monitoring Progress and Modeling with Mathematics<\/p>\n<p>In Exercises 3-6, copy and complete the statement with &lt;, &gt;, or = Explain your reasoning.<\/p>\n<p>Question 3.<br \/>\nm\u22201 ________ m\u22202<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43270\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-141.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 141\" width=\"153\" height=\"97\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49071\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-3.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.6 Question 3\" width=\"564\" height=\"109\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-3.png 564w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-3-300x58.png 300w\" sizes=\"(max-width: 564px) 100vw, 564px\" \/><\/p>\n<p>Question 4.<br \/>\nm\u22201 ________ m\u22202<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43269\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-142.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 142\" width=\"192\" height=\"93\" \/><\/p>\n<p>Answer:<br \/>\nm\u22201 &lt; m\u22202 by the converse of the hinge theorem.<\/p>\n<p>Question 5.<br \/>\nm\u22201 ________ m\u22202<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43268\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-143.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 143\" width=\"170\" height=\"94\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49072\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-5.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.6 Question 5\" width=\"534\" height=\"87\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-5.png 534w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-5-300x49.png 300w\" sizes=\"(max-width: 534px) 100vw, 534px\" \/><\/p>\n<p>Question 6.<br \/>\nm\u22201 ________ m\u22202<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43267\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-144.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 144\" width=\"182\" height=\"92\" \/><\/p>\n<p>Answer:<br \/>\nm\u22201 &gt; m\u22202 by the converse of the hinge theorem.<\/p>\n<p>In Exercises 7-10. copy and complete the statement with &lt;, &gt;, or =. Explain your reasoning.<\/p>\n<p>Question 7.<br \/>\nAD ________ CD<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43266\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-145.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 145\" width=\"151\" height=\"127\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49073\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-7.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.6 Question 7\" width=\"588\" height=\"95\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-7.png 588w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-7-300x48.png 300w\" sizes=\"(max-width: 588px) 100vw, 588px\" \/><\/p>\n<p>Question 8.<br \/>\nMN ________ LK<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43265\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-146.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 146\" width=\"174\" height=\"115\" \/><\/p>\n<p>Answer:<br \/>\nMN &lt; KL by the converse of the hinge theorem.<\/p>\n<p>Question 9.<br \/>\nTR ________ UR<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43264\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-147.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 147\" width=\"170\" height=\"114\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49074\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-9.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.6 Question 9\" width=\"578\" height=\"91\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-9.png 578w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-9-300x47.png 300w\" sizes=\"(max-width: 578px) 100vw, 578px\" \/><\/p>\n<p>Question 10.<br \/>\nAC ________ DC<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43263\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-148.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 148\" width=\"109\" height=\"112\" \/><\/p>\n<p>Answer:<br \/>\nAC &gt; Dc by the converse of the hinge theorem.<\/p>\n<p><strong>PROOF<\/strong><br \/>\nIn Exercises 11 and 12, write a proof.<\/p>\n<p>Question 11.<br \/>\nGiven \\(\\overline{X Y} \\cong \\overline{Y Z}\\), m\u2220WYZ &gt; m\u2220WYX<br \/>\nProve WZ &gt; WX<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43262\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-149.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 149\" width=\"125\" height=\"132\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49075\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-11.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.6 Question 11\" width=\"586\" height=\"427\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-11.png 586w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-11-300x219.png 300w\" sizes=\"(max-width: 586px) 100vw, 586px\" \/><\/p>\n<p>Question 12.<br \/>\nGiven \\(\\overline{B C} \\cong \\overline{D A}\\), DC &lt; AB<br \/>\nProve m\u2220BCA &gt; m\u2220DAC<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43261\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-150.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 150\" width=\"119\" height=\"127\" \/><\/p>\n<p>Answer:<br \/>\nGiven that, AD \u2245 BC<br \/>\nAC \u2245 AC by the reflexive property of congruence theorem<br \/>\nDC &lt; AB is given<br \/>\nSo, m\u2220BCA &gt; m\u2220DAC by the hinge theorem.<\/p>\n<p>In Exercises 13 and 14, you and your friend leave on different flights from the same airport. Determine which flight is farther from the airport. Explain your reasoning.<\/p>\n<p>Question 13.<br \/>\nYour flight: Flies 100 miles due west, then turns 20\u00b0 toward north and flies 50 miles.<br \/>\nFriend\u2019s flight: Flies 100 miles due north, then turns 30\u00b0 toward east and flies 50 miles.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49076\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-13.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.6 Question 13\" width=\"578\" height=\"374\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-13.png 578w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-13-300x194.png 300w\" sizes=\"(max-width: 578px) 100vw, 578px\" \/><\/p>\n<p>Question 14.<br \/>\nYour flight: Flies 21omiles due south, then turns 70\u00b0 toward west and flies 80 miles.<br \/>\nFriend\u2019s flight: Flies 80 miles due north, then turns 50\u00b0 toward east and flies 210 miles.<\/p>\n<p>Answer:<br \/>\nMy friend flight is farther from the airport.<\/p>\n<p>Explanation:<br \/>\nThe figure below represents the situation.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-107021\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-answers-geometry-chapter-6-relationships-within-triangles-6.6-1.jpg\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 6.6 1\" width=\"365\" height=\"690\" \/><br \/>\nI flew 210 miles north to point B then runed 70\u00b0 to the west and flew 80 miles to point C.<br \/>\nMy friend flew 80 miles north to point D then turned 50\u00b0 east and flew 210 miles to point F.<br \/>\nWe should take into consideration that 70\u00b0 to west means 110\u00b0 and 50\u00b0 to east means 130\u00b0<br \/>\nAB = DF<br \/>\nBC = AD<br \/>\nm\u2220ABC &lt; m\u2220ADF<br \/>\nSo, AC &lt; AF<br \/>\nMy friend flight is farther from the airport.<\/p>\n<p>Question 15.<br \/>\n<strong>ERROR ANALYSIS<\/strong><br \/>\nDescribe and correct the error in using the Hinge Theorem (Theorem 6.12).<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43260\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-151.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 151\" width=\"311\" height=\"177\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-151.png 311w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-151-300x171.png 300w\" sizes=\"(max-width: 311px) 100vw, 311px\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49077\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-15.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.6 Question 15\" width=\"553\" height=\"84\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-15.png 553w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-15-300x46.png 300w\" sizes=\"(max-width: 553px) 100vw, 553px\" \/><\/p>\n<p>Question 16.<br \/>\n<strong>REPEATED REASONING<\/strong><br \/>\nWhich is possible measure for \u2220JKM? Select all that apply.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43259\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-152.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 152\" width=\"151\" height=\"128\" \/><br \/>\n(A) 15\u00b0<br \/>\n(B) 22\u00b0<br \/>\n(C) 25\u00b0<br \/>\n(D) 35\u00b0<\/p>\n<p>Answer:<br \/>\n(D) 35\u00b0<\/p>\n<p>Explanation:<br \/>\nKL \u2245 KJ is given<br \/>\nKM \u2245 KM by the reflexive property of congruence<br \/>\nLM &gt; MJ<br \/>\nSo, m\u2220LKM &gt; m\u2220MKJ<br \/>\n25 &gt; 35<\/p>\n<p>Question 17.<br \/>\n<strong>DRAWING CONCLUSIONS<\/strong><br \/>\nThe path from E to F is longer than the path from E to D. The path from G to D is the same length as the path from G to F. What can you conclude about the angles of the paths? Explain your reasoning.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43258\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-153.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 153\" width=\"254\" height=\"105\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49078\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-17.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.6 Question 17\" width=\"585\" height=\"81\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-17.png 585w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-17-300x42.png 300w\" sizes=\"(max-width: 585px) 100vw, 585px\" \/><\/p>\n<p>Question 18.<br \/>\n<strong>ABSTRACT REASONING<\/strong><br \/>\nIn \u2206EFG, the bisector of \u2220F intersects the bisector of \u2220G at point H. Explain why \\(\\overline{F G}\\) must be longer than \\(\\overline{F H}\\) or \\(\\overline{H G}\\).<\/p>\n<p>Answer:<br \/>\nThe sum of interior angles of a triangle is 180 degrees<br \/>\nThe sum of angles F and G cannot be larger than 180 degrees<br \/>\n\u2220HFG is half of the measurement of \u2220F. \u2220HGF is half of the measurement of \u2220G. thus, \u2220HFG + \u2220HGF &lt; 90\u00b0<br \/>\n\u2220FHG = 180\u00b0 &#8211; \u2220HGF &#8211; \u2220HFG &gt; 90\u00b0<\/p>\n<p>Question 19.<br \/>\n<strong>ABSTRACT REASONING<\/strong><br \/>\n\\(\\overline{N R}\\) is a median of \u2206NPQ, and NQ &gt; NP Explain why \u2220NRQ is obtuse.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49079\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-19.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.6 Question 19\" width=\"519\" height=\"172\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-19.png 519w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-19-300x99.png 300w\" sizes=\"(max-width: 519px) 100vw, 519px\" \/><\/p>\n<p><strong>MATHEMATICAL CONNECTIONS<\/strong><br \/>\nIn Exercises 20 and 21, write and solve an inequality for the possible values of x.<\/p>\n<p>Question 20.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43257\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-154.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 154\" width=\"161\" height=\"89\" \/><br \/>\nAnswer:<br \/>\n3x + 2 = x + 3<br \/>\n3x &#8211; x = 3 &#8211; 2<br \/>\n2x = 1<br \/>\nx = \\(\\frac { 1 }{ 2 } \\)<\/p>\n<p>Question 21.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43256\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-155.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 155\" width=\"190\" height=\"81\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49080\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-21.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.6 Question 21\" width=\"594\" height=\"200\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-21.png 594w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-21-300x101.png 300w\" sizes=\"(max-width: 594px) 100vw, 594px\" \/><\/p>\n<p>Question 22.<br \/>\n<strong>HOW DO YOU SEE IT?<\/strong><br \/>\nIn the diagram, triangles are formed by the locations of the players on the basketball court. The dashed lines represent the possible paths of the basketball as the players pass. How does m\u2220ACB compare with m\u2220ACD?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43255\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-156.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 156\" width=\"369\" height=\"199\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-156.png 369w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-156-300x162.png 300w\" sizes=\"(max-width: 369px) 100vw, 369px\" \/><\/p>\n<p>Answer:<br \/>\nBC \u2245 CD<br \/>\nAC \u2245 AC by the reflexive property of congruence theorem<br \/>\nAB &lt; AD<br \/>\nSo, m\u2220ACB &lt; m\u2220ACD by the converse of the hinge theorem.<\/p>\n<p>Question 23.<br \/>\n<strong>CRITICAL THINKING<\/strong><br \/>\nIn \u2206ABC, the altitudes from B and C meet at point D, and m\u2220BDC. What is true about \u2206ABC? Justify your answer.<br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49081\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-23.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.6 Question 23\" width=\"567\" height=\"141\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-23.png 567w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-23-300x75.png 300w\" sizes=\"(max-width: 567px) 100vw, 567px\" \/><\/p>\n<p>Question 24.<br \/>\n<strong>THOUGHT PROVOKING<\/strong><br \/>\nThe postulates and theorems in this book represent Euclidean geometry. In spherical geometry, all points are on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, state an inequality involving the sum of the angles of a triangle. Find a formula for the area of a triangle in spherical geometry.<\/p>\n<p>Answer:<\/p>\n<p>Maintaining Mathematical proficiency<\/p>\n<p>Find the value of x.<\/p>\n<p>Question 25.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43254\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-157.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 157\" width=\"143\" height=\"130\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49082\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-25.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.6 Question 25\" width=\"244\" height=\"101\" \/><\/p>\n<p>Question 26.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43253\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-158.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 158\" width=\"108\" height=\"151\" \/><\/p>\n<p>Answer:<br \/>\nAs two sides are congruent, two angles opposite to the congruent sides are also congruent by the converse of the isosceles trinagle theorem.<br \/>\n36 + x + x = 180<br \/>\n36 + 2x = 180<br \/>\n2x = 144<br \/>\nx = 72<\/p>\n<p>Question 27.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43252\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-159.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 159\" width=\"138\" height=\"122\" \/><br \/>\nAnswer:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-49083\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-6.6-Question-27.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 6.6 Question 27\" width=\"122\" height=\"65\" \/><\/p>\n<p>Question 28.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43251\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-160.png\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 160\" width=\"159\" height=\"158\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-160.png 159w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6-Relationships-Within-Triangles-160-150x150.png 150w\" sizes=\"(max-width: 159px) 100vw, 159px\" \/><br \/>\nAnswer:<br \/>\nThe sum of interior angles in\u00a0 a triangle are 180 degrees<br \/>\n44 + 64 + y = 180<br \/>\ny = 180 &#8211; 108<br \/>\ny = 72<br \/>\nx + y = 180<br \/>\nx + 72 = 180<br \/>\nx = 108<\/p>\n<h3><a id=\"Relationships_Within_Triangles_Chapter_Review\"><\/a>Relationships Within Triangles Chapter Review<\/h3>\n<h3>6.1 Perpendicular and Angle Bisectors<\/h3>\n<p>Find the indicated measure. Explain your reasoning.<\/p>\n<p>Question 1.<br \/>\nDC<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43497\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-161.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 161\" width=\"177\" height=\"113\" \/><\/p>\n<p>Answer:<br \/>\nDC = 20<\/p>\n<p>Explanation:<br \/>\nAD = CD<br \/>\n20 = 7x &#8211; 15<br \/>\n7x = 35<br \/>\nx = 5<br \/>\nDC = 7(5) &#8211; 15 = 20<\/p>\n<p>Question 2.<br \/>\nRS<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43496\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-162.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 162\" width=\"133\" height=\"129\" \/><\/p>\n<p>Answer:<br \/>\nRS = 23<\/p>\n<p>Explanation:<br \/>\nPS = SR<br \/>\n6x + 5 = 9x &#8211; 4<br \/>\n9x &#8211; 6x = 9<br \/>\n3x = 9<br \/>\nx = 3<br \/>\nRS = 9(3) &#8211; 4 = 23<\/p>\n<p>Question 3.<br \/>\nm\u2220JFH<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43495\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-163.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 163\" width=\"133\" height=\"137\" \/><\/p>\n<p>Answer:<br \/>\nm\u2220JFH = 47\u00b0<\/p>\n<p>Explanation:<br \/>\nm\u2220JFH = m\u2220JFG<br \/>\nSo, m\u2220JFH = 47\u00b0<\/p>\n<h3>6.2 Bisectors of Triangles<\/h3>\n<p>Find the coordinates of the circumcenter of the triangle with the given vertices.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-43494\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-164.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 164\" width=\"206\" height=\"128\" \/><\/p>\n<p>Question 4.<br \/>\nT(- 6, &#8211; 5), U(0, &#8211; 1), V(0, &#8211; 5)<\/p>\n<p>Answer:<br \/>\nThe circumcenter is (-3, -3)<\/p>\n<p>Explanation:<br \/>\nRecall that the circumcentre of a triangle is equidistant from the vertices of a triangle. Let T(- 6, &#8211; 5), U(0, &#8211; 1), V(0, &#8211; 5) be the vertices of the given triangle and let P(x,y) be the circumcentre of this triangle. Then<br \/>\nPT = PU = PV<br \/>\nPT\u00b2 = PU\u00b2 = PV\u00b2<br \/>\nPT\u00b2 = PU\u00b2<br \/>\n(x + 6)\u00b2 + (y + 5)\u00b2 = (x &#8211; 0)\u00b2 + (y + 1)\u00b2<br \/>\nx\u00b2 + 12x + 36 + y\u00b2 + 10y + 25 = x\u00b2 + y\u00b2 + 2y + 1<br \/>\n12x + 8y + 61 = 1<br \/>\n12x + 8y + 60 = 0<br \/>\n3x + 2y + 15 = 0<br \/>\nPU\u00b2 = PV\u00b2<br \/>\n(x &#8211; 0)\u00b2 + (y + 1)\u00b2 = (x &#8211; 0)\u00b2 + (y + 5)\u00b2<br \/>\nx\u00b2 + y\u00b2 + 2y + 1 = x\u00b2 + y\u00b2 + 10y + 25<br \/>\n8y + 24 = 0<br \/>\n8y = -24<br \/>\ny = -3<br \/>\nSubstitute y = -3 in 3x + 2y + 15 = 0<br \/>\n3x + 2(-3) + 15 = 0<br \/>\n3x &#8211; 6 + 15 = 0<br \/>\n3x = -9<br \/>\nx = -3<br \/>\nThe circumcenter is (-3, -3)<\/p>\n<p>Question 5.<br \/>\nX(- 2, 1), Y(2, &#8211; 3), Z(6, &#8211; 3)<\/p>\n<p>Answer:<br \/>\nThe circumcenter is (4, 3)<\/p>\n<p>Explanation:<br \/>\nRecall that the circumcentre of a triangle is equidistant from the vertices of a triangle. Let X(- 2, 1), Y(2, &#8211; 3), Z(6, &#8211; 3) be the vertices of the given triangle and let P(x, y) be the circumcentre of this triangle. Then<br \/>\nPX = PY = PZ<br \/>\nPX\u00b2 = PY\u00b2 = PZ\u00b2<br \/>\nPX\u00b2 = PY\u00b2<br \/>\n(x + 2)\u00b2 + (y &#8211; 1)\u00b2 = (x &#8211; 2)\u00b2 + (y + 3)\u00b2<br \/>\nx\u00b2 + 4x + 4 + y\u00b2 &#8211; 2y + 1 = x\u00b2 &#8211; 4x + 4 + y\u00b2 + 6y + 9<br \/>\n8x &#8211; 8y + 5 = 13<br \/>\n8x &#8211; 8y &#8211; 8 = 0<br \/>\nx &#8211; y = 1<br \/>\nPY\u00b2 = PZ\u00b2<br \/>\n(x &#8211; 2)\u00b2 + (y + 3)\u00b2 = (x &#8211; 6)\u00b2 + (y + 3)\u00b2<br \/>\nx\u00b2 &#8211; 4x + 4 + y\u00b2 + 6y + 9 = x\u00b2 &#8211; 12x + 36 + y\u00b2 + 6y + 9<br \/>\n8x = 32<br \/>\nx = 4<br \/>\nSubstitue x = 4 in x &#8211; y = 1<br \/>\n4 &#8211; y = 1<br \/>\n-y = -3<br \/>\ny = 3<br \/>\nThe circumcenter is (4, 3)<\/p>\n<p>Question 6.<br \/>\nPoint D is the incenter of \u2206LMN. Find the value of x.<\/p>\n<p>Answer:<br \/>\nx = 13<\/p>\n<h3>6.3 Medians and Altitudes of Triangles<\/h3>\n<p>Find the coordinates of the centroid of the triangle with the given vertices.<\/p>\n<p>Question 7.<br \/>\nA(- 10, 3), B(- 4, 5), C(- 4, 1)<\/p>\n<p>Answer:<br \/>\nThe centroid is (-6, 3)<\/p>\n<p>Explanation:<br \/>\nThe centroid = (\\(\\frac { -10 &#8211; 4 &#8211; 4 }{ 3 } \\), \\(\\frac { 3 + 5 + 1 }{ 3 } \\))<br \/>\n= (-6, 3)<\/p>\n<p>Question 8.<br \/>\nD(2, &#8211; 8), E(2, &#8211; 2), F(8, &#8211; 2)<\/p>\n<p>Answer:<br \/>\nThe centroid is (4, -4)<\/p>\n<p>Explanation:<br \/>\nThe centroid = (\\(\\frac { 2 + 2 + 8 }{ 3 } \\), \\(\\frac { -8 &#8211; 2 &#8211; 2 }{ 3 } \\))<br \/>\n= (4, -4)<\/p>\n<p>Tell whether the orthocenter of the triangle with the given vertices is inside, on, or outside the triangle. Then find the coordinates of the orthocenter.<\/p>\n<p>Question 9.<br \/>\nG(1, 6), H(5, 6), J(3, 1)<\/p>\n<p>Answer:<br \/>\nThe orthocenter is (3, 5.2)<br \/>\nThe orthocenter lies inside the triangle.<\/p>\n<p>Explanation:<br \/>\nThe slope of the line HJ = \\(\\frac { 1 &#8211; 6 }{ 3 &#8211; 5} \\) = \\(\\frac { 5 }{ 2 } \\)<br \/>\nThe slope of the perpendicular line = \\(\\frac { -2 }{ 5 } \\)<br \/>\nThe perpendicular line is (y &#8211; 6) = \\(\\frac { -2 }{ 5 } \\)(x &#8211; 1)<br \/>\n5(y &#8211; 6) = -2(x &#8211; 1)<br \/>\n5y &#8211; 30 = -2x + 2<br \/>\n2x + 5y &#8211; 32 = 0 &#8212; (i)<br \/>\nThe slope of GJ = \\(\\frac { 1 &#8211; 6 }{ 3 &#8211; 1 } \\) = \\(\\frac { -5 }{ 2 } \\)<br \/>\nThe slope of the perpendicular line = \\(\\frac { 2 }{ 5 } \\)<br \/>\nThe equation of perpendicular line (y &#8211; 6) = \\(\\frac { 2 }{ 5 } \\)(x &#8211; 5)<br \/>\n5(y &#8211; 6) = 2(x &#8211; 5)<br \/>\n5y &#8211; 30 = 2x &#8211; 10<br \/>\n2x &#8211; 5y + 20 = 0 &#8212; (ii)<br \/>\nEquate both equations<br \/>\n2x + 5y &#8211; 32 = 2x &#8211; 5y + 20<br \/>\n10y = 52<br \/>\ny = 5.2<br \/>\nSubstitute y = 5.2 in (i)<br \/>\n2x + 5(5.2) &#8211; 32 = 0<br \/>\n2x + 26 &#8211; 32 = 0<br \/>\n2x = 6<br \/>\nx = 3<br \/>\nThe orthocenter is (3, 5.2)<br \/>\nThe orthocenter lies inside the triangle.<\/p>\n<p>Question 10.<br \/>\nK(-8, 5), L(- 6, 3), M(0, 5)<\/p>\n<p>Answer:<br \/>\nThe othrocenter is (-6, -1)<br \/>\nThe orthocenter lies outside of the triangle<\/p>\n<p>Explanation:<br \/>\nThe slope of LM = \\(\\frac { 5 &#8211; 3}{ 0 + 6} \\) = \\(\\frac { 1 }{ 3 } \\)<br \/>\nThe slope of the perpendicular line = -3<br \/>\nThe perpendicular line is (y &#8211; 5) = -3(x + 8)<br \/>\ny &#8211; 5 = -3x &#8211; 24<br \/>\n3x + y + 19 = 0 &#8212;- (ii)<br \/>\nThe slope of KL = \\(\\frac { 3 &#8211; 5 }{ -6 + 8 } \\) = -1<br \/>\nThe slope of the perpendicular line = \\(\\frac { 1 }{ 2 } \\)<br \/>\nThe equation of perpendicular line (y &#8211; 5) = \\(\\frac { 1 }{ 2 } \\)(x &#8211; 0)<br \/>\n2y &#8211; 10 = x &#8212;- (ii)<br \/>\nSubstitute (ii) in (i)<br \/>\n3(2y &#8211; 10) + y + 19 = 0<br \/>\n6y &#8211; 30 + y + 19 = 0<br \/>\n7y &#8211; 11 = 0<br \/>\ny = \\(\\frac { 11 }{ 7 } \\)<br \/>\nx = -6<br \/>\nThe othrocenter is (-6, -1)<br \/>\nThe orthocenter lies outside of the triangle<\/p>\n<h3>6.4 The Triangle Midsegment Theorem<\/h3>\n<p>Find the coordinates of the vertices of the midsegment triangle for the triangle with the given vertices.<\/p>\n<p>Question 11.<br \/>\nA(- 6, 8), B(- 6, 4), C(0, 4)<\/p>\n<p>Answer:<br \/>\nThe midsegment of AB = (-6, 6)<br \/>\nThe midsegment of BC = (-3, 4)<br \/>\nThe midsegment of AC = (-3, 6)<\/p>\n<p>Explanation:<br \/>\nThe midsegment of AB = (\\(\\frac { -6 &#8211; 6 }{ 2 } \\), \\(\\frac { 8 + 4 }{ 2 } \\))<br \/>\n= (-6, 6)<br \/>\nThe midsegment of BC = (\\(\\frac { -6 + 0 }{ 2 } \\), \\(\\frac { 4 + 4 }{ 2 } \\))<br \/>\n= (-3, 4)<br \/>\nThe midsegment of AC = (\\(\\frac { -6 + 0 }{ 2 } \\), \\(\\frac { 8 + 4 }{ 2 } \\))<br \/>\n= (-3, 6)<\/p>\n<p>Question 12.<br \/>\nD(- 3, 1), E(3, 5), F(1, &#8211; 5)<\/p>\n<p>Answer:<br \/>\nThe midsegment of DE = (0, 3)<br \/>\nThe midsegment of EF = (2, 0)<br \/>\nThe midsegment of DF = (-1, -2)<\/p>\n<p>Explanation:<br \/>\nThe midsegment of DE = (\\(\\frac { -3 + 3 }{ 2 } \\), \\(\\frac { 1 + 5 }{ 2 } \\))<br \/>\n= (0, 3)<br \/>\nThe midsegment of EF = (\\(\\frac { 3 + 1 }{ 2 } \\), \\(\\frac { 5 &#8211; 5 }{ 2 } \\))<br \/>\n= (2, 0)<br \/>\nThe midsegment of DF = (\\(\\frac { -3 + 1 }{ 2 } \\), \\(\\frac { 1 &#8211; 5 }{ 2 } \\))<br \/>\n= (-1, -2)<\/p>\n<h3>6.5 Indirect Proof and Inequalities in One Triangle<\/h3>\n<p>Describe the possible lengths of the third side of the triangle given the lengths of the other two sides.<\/p>\n<p>Question 13.<br \/>\n4 inches, 8 inches<\/p>\n<p>Answer:<br \/>\n4 &lt; x &lt; 12<\/p>\n<p>Explanation:<br \/>\n4 + 8 &gt; x<br \/>\n12 &gt; x<br \/>\n4 + x &gt; 8<br \/>\nx &gt; 4<br \/>\n8 + x &gt; 4<br \/>\nx &gt; -4<br \/>\n4 &lt; x &lt; 12<\/p>\n<p>Question 14.<br \/>\n6 meters, 9 meters<\/p>\n<p>Answer:<br \/>\n3 &lt; x &lt; 15<\/p>\n<p>Explanation:<br \/>\n6 + 9 &gt; x<br \/>\n15 &gt; x<br \/>\n6 + x &gt; 9<br \/>\nx &gt; 3<br \/>\n9 + x &gt; 6<br \/>\nx &gt; -3<br \/>\n3 &lt; x &lt; 15<\/p>\n<p>Question 15.<br \/>\n11 feet, 18 feet<\/p>\n<p>Answer:<br \/>\n7 &lt; x &lt; 29<\/p>\n<p>Explanation:<br \/>\n11 + 18 &gt; x<br \/>\n29 &gt; x<br \/>\n11 + x &gt; 18<br \/>\nx &gt; 7<br \/>\n18 + x &gt; 11<br \/>\nx &gt; -7<br \/>\n7 &lt; x &lt; 29<\/p>\n<p>Question 16.<br \/>\nWrite an indirect proof 0f the statement &#8220;In \u2206XYZ, if XY = 4 and XZ = 8. then YZ &gt; 4.\u201d<\/p>\n<p>Answer:<br \/>\n4 + 8 &gt; x<br \/>\n12 &gt; x<br \/>\n4 + x &gt; 8<br \/>\nx &gt; 4<br \/>\n8 + x &gt; 4<br \/>\nx &gt; -4<br \/>\nSo, 12 &gt; x &gt; 4<br \/>\nTherefore, YZ &gt; 4<\/p>\n<h3>6.6 Inequalities in Two Triangles<\/h3>\n<p>Use the diagram.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-43493\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-165.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 165\" width=\"149\" height=\"125\" \/><\/p>\n<p>Question 17.<br \/>\nIf RQ = RS and m\u2220QRT &gt; m\u2220SRT, then how does \\(\\overline{Q T}\\) Compare to \\(\\overline{S T}\\)?<\/p>\n<p>Answer:<br \/>\nTQ &gt; ST<\/p>\n<p>Explanation:<br \/>\nGiven that,<br \/>\nRQ = RS and m\u2220QRT &gt; m\u2220SRT<br \/>\nRT = RT by the reflexive property of congruence theorem<br \/>\nTQ &gt; ST by the converse of the hinge theorem.<\/p>\n<p>Question 18.<br \/>\nIf RQ = RS and QT &gt; ST, then how does \u2220QRT compare to \u2220SRT?<\/p>\n<p>Answer:<br \/>\nGiven that,<br \/>\nRQ = RS and QT &gt; ST<br \/>\nRT = RT by the reflexive property of congruence theorem<br \/>\nSo, m\u2220QRT &gt; m\u2220SRT by the converse of the hinge theorem.<\/p>\n<h3><a id=\"Relationships_Within_Triangles_Chapter_Test\"><\/a>Relationships Within Triangles Chapter Test<\/h3>\n<p>In Exercise 1 and 2, \\(\\overline{M N}\\) is a midsegment of \u2206JKL. Find the value of x.<\/p>\n<p>Question 1.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43511\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-166.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 166\" width=\"131\" height=\"76\" \/><\/p>\n<p>Answer:<br \/>\nx = 6<\/p>\n<p>Explanation:<br \/>\nKL = 0.5(MN)<br \/>\n12 = 0.5x<br \/>\nx = 6<\/p>\n<p>Question 2.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43510\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-167.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 167\" width=\"92\" height=\"104\" \/><\/p>\n<p>Answer:<br \/>\nx = 9<\/p>\n<p>Explanation:<br \/>\nJM = ML<br \/>\nx = 9<\/p>\n<p>Find the indicated measure. Identify the theorem you use.<\/p>\n<p>Question 3.<br \/>\nST<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43509\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-168.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 168\" width=\"153\" height=\"137\" \/><\/p>\n<p>Answer:<br \/>\nST = 17<\/p>\n<p>Explanation:<br \/>\nRS = ST by the perpendicular bisector theorem<br \/>\n3x + 8 = 7x &#8211; 4<br \/>\n7x &#8211; 3x = 12<br \/>\n4x = 12<br \/>\nx = 3<br \/>\nST = 7(3) &#8211; 4 = 21 &#8211; 4 = 17<\/p>\n<p>Question 4.<br \/>\nWY<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43508\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-169.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 169\" width=\"162\" height=\"103\" \/><\/p>\n<p>Answer:<br \/>\nWY = 32<\/p>\n<p>Explanation:<br \/>\n6x + 2 = 9x &#8211; 13 by the angle bisector theorem<br \/>\n3x = 15<br \/>\nx = 5<br \/>\nWY = 6(5) + 2 = 32<\/p>\n<p>Question 5.<br \/>\nBW<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43507\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-170.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 170\" width=\"237\" height=\"130\" \/><\/p>\n<p>Answer:<br \/>\nBW = 20<\/p>\n<p>Explanation:<br \/>\nWC = BW = AW by the incenter theorem<br \/>\nBW = 20<\/p>\n<p>Copy and complete the statement with &lt;, &gt;, or =.<\/p>\n<p>Question 6.<br \/>\nAB _____ CB<\/p>\n<p>Answer:<br \/>\nAB &gt; CB<\/p>\n<p>Question 7.<br \/>\nm\u22201 _____ m\u22202<\/p>\n<p>Answer:<br \/>\nm\u22201 &lt; m\u22202 by the converse of the hinge theorem<\/p>\n<p>Question 8.<br \/>\nm\u2220MNP ________ m\u2220NPM<\/p>\n<p>Answer:<br \/>\nm\u2220MNP &lt; m\u2220NPM<\/p>\n<p>Question 9.<br \/>\nFind the coordinates of the circumcenter, orthocenter, and centroid of the triangle with vertices A(0, &#8211; 2), B(4, &#8211; 2), and C(0, 6).<\/p>\n<p>Answer:<br \/>\nCircumcenter is (1.5, -0.5)<br \/>\northoceter is (0, -2)<br \/>\ncentroid is (1.3, 0.7)<\/p>\n<p>Explanation:<br \/>\nGraph the triangle<br \/>\nDraw the bisector of every angle and their intersection to get the circumcenter<br \/>\nDraw the altitudes and their intersections to get orthocenter<br \/>\nFind the midpoint of all sides and then connect it with the opposite vertex. The intersection of these medians is the centroid.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-107388\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/big-ideas-math-geometry-answers-chapter-6-relationships-within-triangles-1.jpg\" alt=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles 1\" width=\"630\" height=\"631\" \/><br \/>\nCircumcenter is (1.5, -0.5)<br \/>\northoceter is (0, -2)<br \/>\ncentroid is (1.3, 0.7)<\/p>\n<p>Question 10.<br \/>\nWrite an indirect proof of the Corollary to the Base Angles Theorem (Corollary 5.2): If \u2206PQR is equilateral, then it is equiangular.<\/p>\n<p>Answer:<br \/>\nAssume for contradiction that there is an equilateral triangle namely \u25b3ABC. So, AB = BC = AC. According to the base angles theorem, if two sides of a triangle are congruent, then the angles opposite them are congruent.<br \/>\nSince AB = AC, the base angle theorem tells that m\u2220B = m\u2220C<br \/>\nSince AB = BC, the base angle theorem tells that m\u2220A = m\u2220C<br \/>\nBy the transitive property of equality m\u2220A = m\u2220B = m\u2220C, which contradicts the assumption that triangle ABC is equiangular. Thus, the corollary is true.<\/p>\n<p>Question 11.<br \/>\n\u2206DEF is a right triangle with area A. Use the area for \u2206DEF to write an expression for the area of \u2206GEH. Justify your answer.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43506\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-171.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 171\" width=\"93\" height=\"87\" \/><\/p>\n<p>Answer:<br \/>\nThe area of \u2206DEF = A<br \/>\nAs HG is the midsegment of \u2206DEF.<br \/>\nArea of \u2206GEH = A\/2<\/p>\n<p>Question 12.<br \/>\nTwo hikers start at a visitor center. The first hikes 4 miles hikes due west, then turns 40\u00b0 toward south and hikes 1.8 miles. The second hikes 4 miles due east, then turns 52\u00b0 toward north and hikes 1 .8 miles. Which hiker is farther from the visitor center? Explain how you know.<\/p>\n<p>Answer:<br \/>\nThe second hiker is farther away from the visitor center as all things being equal cos 128 &lt; cos 140 from distance\u00b2 = 4\u00b2 + 1.8\u00b2 &#8212; 2 x 4 x 1.8 cos (angle turned)<\/p>\n<p>In Exercises 13-15, use the map.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-43505\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-172.png\" alt=\"Big Ideas Math Answers Geometry Chapter 6 Relationships Within Triangles 172\" width=\"307\" height=\"195\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-172.png 307w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-6-Relationships-Within-Triangles-172-300x191.png 300w\" sizes=\"(max-width: 307px) 100vw, 307px\" \/><\/p>\n<p>Question 13.<br \/>\nDescribe the possible lengths of Pine Avenue.<\/p>\n<p>Answer:<br \/>\nThe possible lengths of pine avenue are between 2 and 16.<\/p>\n<p>Explanation:<br \/>\n7 + 9 &gt; x<br \/>\n16 &gt; x<br \/>\n7 + x &gt; 9<br \/>\nx &gt; 2<br \/>\n9 + x &gt; 7<br \/>\nx &gt; -2<br \/>\n2 &lt; x &lt; 16<br \/>\nThe possible lengths of pine avenue are between 2 and 16.<\/p>\n<p>Question 14.<br \/>\nYou ride your bike along a trail that represents the shortest distance from the beach to Main Street. You end up exactly halfway between your house and the movie theatre. How long is Pine Avenue? Explain.<\/p>\n<p>Answer:<br \/>\nhalfway between your house and the movie theatre = 7\/2 = 3.5<br \/>\nPine Avenue is 3.5 mi<\/p>\n<p>Question 15.<br \/>\nA market is the same distance from your house, the movie theater, and the beach. Copy the map and locate the market.<\/p>\n<p>Answer:<br \/>\nCentroid is the market.<\/p>\n<h3><a id=\"Relationships_Within_Triangles_Cumulative_Assessment\"><\/a>Relationships Within Triangles Cumulative Assessment<\/h3>\n<p>Question 1.<br \/>\nWhich definitions(s) and\/or theorem(s) do you need to use to prove the Converse of the Perpendicular Bisector Theorem (Theorem 6.2)? Select all that apply.<br \/>\nGiven CA = CB<br \/>\nProve Point C lies on the perpendicular bisector of AB.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43550\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-173.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 173\" width=\"169\" height=\"108\" \/><\/p>\n<table border=\"2\">\n<tbody>\n<tr>\n<td width=\"312\">Definition of perpendicular bisector<\/td>\n<td width=\"312\">Definition of angle bisector<\/td>\n<\/tr>\n<tr>\n<td width=\"312\">Definition of segment congruence<\/td>\n<td width=\"312\">Definition of angle congruence<\/td>\n<\/tr>\n<tr>\n<td width=\"312\">Base Angles Theorem (Theorem 5.6)<\/td>\n<td width=\"312\">Converse of the Base Angles Theorem (Theorem 5.7)<\/td>\n<\/tr>\n<tr>\n<td width=\"312\">ASA Congruence Theorem (Theorem 5.10)<\/td>\n<td width=\"312\">AAS Congruence Theorem (Theorem 5.11)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Answer:<br \/>\nDefinition of perpendicular bisector is used to prove the Converse of the Perpendicular Bisector Theorem.<\/p>\n<p>Question 2.<br \/>\nUse the given information to write a two-column proof.<br \/>\nGiven \\(\\overline{Y G}\\) is the perpendicular bisector of \\(\\overline{D F}\\).<br \/>\nProve \u2206DEY \u2245 \u2206FEY<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43549\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-174.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 174\" width=\"167\" height=\"169\" \/><br \/>\nAnswer:<\/p>\n<p>Question 3.<br \/>\nWhat are the coordinates of the centroid of \u2206LMN?<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43548\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-175.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 175\" width=\"174\" height=\"177\" \/><br \/>\n(A) (2, 5)<br \/>\n(B) (3, 5)<br \/>\n(C) (4, 5)<br \/>\n(D) (5, 5)<\/p>\n<p>Answer:<br \/>\n(B) (3, 5)<\/p>\n<p>Explanation:<br \/>\nThe coordinates of L(3, 8), M (1, 5), N(5, 2)<br \/>\nCentroid = (\\(\\frac { 3 + 1 + 5 }{ 3 } \\), \\(\\frac { 8 + 5 + 2 }{ 3 } \\))<br \/>\n= (3, 5)<\/p>\n<p>Question 4.<br \/>\nUse the steps in the construction to explain how you know that the circle is circumscribed about \u2206ABC.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43547\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-176.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 176\" width=\"702\" height=\"232\" srcset=\"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-176.png 702w, https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-176-300x99.png 300w\" sizes=\"(max-width: 702px) 100vw, 702px\" \/><\/p>\n<p>Answer:<br \/>\nDraw a perpendicular bisector for AB<br \/>\nDraw a perpendicular bisector for BC<br \/>\nExtend those bisectors to meet at one point.<br \/>\nBy taking the point of intersection of perpendicular bisectors as radius, draw a circle that connects three vertices of the triangle.<\/p>\n<p>Question 5.<br \/>\nEnter the missing reasons in the proof of the Base Angles Theorem (Theorem 5.6).<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43546\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-177.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 177\" width=\"199\" height=\"123\" \/><br \/>\nGiven \\(\\overline{A B} \\cong \\overline{\\Lambda C}\\)<br \/>\nProve \u2220B = \u2220C<\/p>\n<table border=\"2\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\" width=\"312\">Statements<\/td>\n<td style=\"text-align: center;\" width=\"312\">Reasons<\/td>\n<\/tr>\n<tr>\n<td width=\"312\">1. Draw \\(\\overline{A D}\\), the angle bisector of \u2220CAB<\/td>\n<td width=\"312\">1. Construction of angle bisector<\/td>\n<\/tr>\n<tr>\n<td width=\"312\">2. \u2220CAD \u2245 \u2220BAD<\/td>\n<td width=\"312\">2. ________________________<\/td>\n<\/tr>\n<tr>\n<td width=\"312\">3. \\( \\overline{\\Lambda B} \\cong \\overline{A C} \\)<\/td>\n<td width=\"312\">3. ________________________<\/td>\n<\/tr>\n<tr>\n<td width=\"312\">4. \\( \\overline{D A} \\cong \\overline{D A} \\)<\/td>\n<td width=\"312\">4. ________________________<\/td>\n<\/tr>\n<tr>\n<td width=\"312\">5. \u2206ADB \u2245 \u2206ADC<\/td>\n<td width=\"312\">5. ________________________<\/td>\n<\/tr>\n<tr>\n<td width=\"312\">6. \u2220B \u2245 \u2220C<\/td>\n<td width=\"312\">6. ________________________<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Answer:<\/p>\n<table border=\"2\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\" width=\"312\">Statements<\/td>\n<td style=\"text-align: center;\" width=\"312\">Reasons<\/td>\n<\/tr>\n<tr>\n<td width=\"312\">1. Draw \\(\\overline{A D}\\), the angle bisector of \u2220CAB<\/td>\n<td width=\"312\">1. Construction of angle bisector<\/td>\n<\/tr>\n<tr>\n<td width=\"312\">2. \u2220CAD \u2245 \u2220BAD<\/td>\n<td width=\"312\">2. Angle Bisector Congruence Theorem<\/td>\n<\/tr>\n<tr>\n<td width=\"312\">3. \\( \\overline{\\Lambda B} \\cong \\overline{A C} \\)<\/td>\n<td width=\"312\">3. Given<\/td>\n<\/tr>\n<tr>\n<td width=\"312\">4. \\( \\overline{D A} \\cong \\overline{D A} \\)<\/td>\n<td width=\"312\">4. Reflexive property of congruence theorem<\/td>\n<\/tr>\n<tr>\n<td width=\"312\">5. \u2206ADB \u2245 \u2206ADC<\/td>\n<td width=\"312\">5. SAS congruence theorem<\/td>\n<\/tr>\n<tr>\n<td width=\"312\">6. \u2220B \u2245 \u2220C<\/td>\n<td width=\"312\">6. Angle congruence theorem<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Question 6.<br \/>\nUse the graph of \u2206QRS.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43545\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-178.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 178\" width=\"172\" height=\"168\" \/><br \/>\na. Find the coordinates of the vertices of the midsegment triangle. Label the vertices T, U, and V.<\/p>\n<p>Answer:<br \/>\nT(0, 7), U(2, 4), V(-1, 5)<\/p>\n<p>Explanation:<br \/>\nThe coordinates of Q(-3, 8), R(3, 6), S(1, 2)<br \/>\nThe midpoint of QR = T = (\\(\\frac { -3 +3 }{ 2 } \\), \\(\\frac { 8 + 6 }{ 2 } \\)) = (0, 7)<br \/>\nMidpoint of RS = U = (\\(\\frac { 3 + 1 }{ 2 } \\), \\(\\frac { 6 + 2 }{ 2 } \\)) = (2, 4)<br \/>\nMidpoint of SR = V = (\\(\\frac { 1 &#8211; 3 }{ 2 } \\), \\(\\frac { 2 + 8 }{ 2 } \\)) = (-1, 5)<\/p>\n<p>b. Show that each midsegment joining the midpoints of two sides is parallel to the third side and is equal to half the length of the third side.<br \/>\nAnswer:<br \/>\nThe slope of RS = \\(\\frac { 2 &#8211; 6 }{ 1 &#8211; 3 } \\) = 2<br \/>\nThe slope of TV = \\(\\frac { 5 &#8211; 7 }{ -1 &#8211; 0 } \\) = 2<br \/>\nAs the slopes are same. The lines are parallel.<br \/>\nTV = \u221a(-1 &#8211; 0)\u00b2 + (5 &#8211; 7)\u00b2 = \u221a1 + 4 = \u221a5<br \/>\nRS = \u221a(1 &#8211; 3)\u00b2 + (2 &#8211; 6)\u00b2 = \u221a4 + 16 = \u221a20<br \/>\nTV = 0.5(RS)<\/p>\n<p>Question 7.<br \/>\nA triangle has vertices X(- 2, 2), Y(1, 4), and Z(2, &#8211; 2). Your friend claims that a translation of (x, y) \u2192 (x + 2, y &#8211; 3) and a translation by a scale factor of 3 will produce a similarity transformation. Do you support our friend\u2019s claim? Explain our reasoning.<\/p>\n<p>Answer:<\/p>\n<p>Question 8.<br \/>\nThe graph shows a dilation of quadrilateral ABCD b a scale factor of 2. Show that the line containing points B and D is parallel to the line Containing points B&#8217; and D&#8217;.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-43544\" src=\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-6-Relationships-Within-Triangles-179.png\" alt=\"Big Ideas Math Geometry Answer Key Chapter 6 Relationships Within Triangles 179\" width=\"208\" height=\"144\" \/><br \/>\nAnswer:<br \/>\nThe coordinates of B(4, 1), D(1, -1), B'(8, 2), D'(2, -2)<br \/>\nThe slope of BD = \\(\\frac { -1 &#8211; 1 }{ 1 &#8211; 4 } \\) = \\(\\frac { 2 }{ 3 } \\)<br \/>\nThe slope of B&#8217;D&#8217; = \\(\\frac { -2 &#8211; 2 }{ 2 &#8211; 8 } \\) = \\(\\frac { -4 }{ -6 } \\) = \\(\\frac { 2 }{ 3 } \\)<br \/>\nThe slopes are equal. So, BD is parallel to B&#8217;D&#8217;.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Give your kid the right assistance he\/she needs regarding the Big Ideas Math Geometry Chapter 6 Relationships within Triangles concepts. Practicing from the Big Ideas Math Geometry Answers Ch 6 Relationships within Triangles on a frequent basis help you to attempt the exam with confidence. You can avail these handy resources for BIM Book Geometry &#8230; <a title=\"Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles\" class=\"read-more\" href=\"https:\/\/bigideasmathanswers.com\/big-ideas-math-geometry-answers-chapter-6\/\" aria-label=\"More on Big Ideas Math Geometry Answers Chapter 6 Relationships Within Triangles\">Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":89491,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[4],"tags":[],"jetpack_featured_media_url":"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-6.png","_links":{"self":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/41131"}],"collection":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/comments?post=41131"}],"version-history":[{"count":1,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/41131\/revisions"}],"predecessor-version":[{"id":90290,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/41131\/revisions\/90290"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media\/89491"}],"wp:attachment":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media?parent=41131"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/categories?post=41131"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/tags?post=41131"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}