{"id":37019,"date":"2023-10-24T10:21:47","date_gmt":"2023-10-24T04:51:47","guid":{"rendered":"https:\/\/ccssmathanswers.com\/?p=37019"},"modified":"2023-11-10T12:01:06","modified_gmt":"2023-11-10T06:31:06","slug":"properties-of-a-rectangle-rhombus-and-square","status":"publish","type":"post","link":"https:\/\/bigideasmathanswers.com\/properties-of-a-rectangle-rhombus-and-square\/","title":{"rendered":"Properties of a Rectangle Rhombus and Square | Special Parallelograms Properties"},"content":{"rendered":"<p>Properties of a Rectangle Rhombus and Square is always a confusing concept for students. Learning every individual topic is important to score good marks in the exam. So, we have explained every individual topic clearly in a detailed manner in this article. Therefore, those who wish to learn the concepts of Parallelogram and its properties, problems, can completely learn the <a href=\"https:\/\/ccssmathanswers.com\/parallelogram\/\">Parallelogram<\/a> concepts on our website.<\/p>\n<h3>Rectangle<\/h3>\n<p>A rectangle is said to be a parallelogram when it has all 4 angles having equal measure.<\/p>\n<h4>Properties of Rectangle<\/h4>\n<ul>\n<li>The Opposite sides of a rectangle are parallel.<\/li>\n<li>Also, the Opposite sides of a rectangle are equal in length.<\/li>\n<li>Diagonals are equal in length.<\/li>\n<li>The interior angles are 90 degrees each.<\/li>\n<li>Diagonals bisect each other.<\/li>\n<li>It has horizontal and vertical lines\u00a0of symmetry.<\/li>\n<li>Each of the diagonal bisects the rectangle into 2 congruent triangles.<\/li>\n<li>If you combine the 4 sides of a rectangle, then the mid-points of it form a rhombus.<\/li>\n<\/ul>\n<h4>Rectangle Formulas<\/h4>\n<p>If l is the length of the rectangle and b is the breadth of the rectangle, then<br \/>\nArea = lb square units<br \/>\nPerimeter = 2 (l+b) units.<\/p>\n<h4>Diagonal Properties of a Rectangle<\/h4>\n<p>Prove that the diagonals of a rectangle are equal and bisect each other.<\/p>\n<p>Proof:<br \/>\nLet PQRS be a rectangle that has diagonals PQ and QS intersect at the point O.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-7270\" src=\"https:\/\/www.worksheetsbuddy.com\/wp-content\/uploads\/2021\/01\/rectangle-300x168.jpg\" alt=\"rectangle\" width=\"229\" height=\"128\" \/><\/p>\n<p>From \u2206 PQR and \u2206 QPS,<br \/>\nPQ = QP (common)<br \/>\n\u2220PQR = \u2220QPS (each equal to 90\u00ba)<br \/>\nQR = PS (opposite sides of a rectangle).<br \/>\nTherefore, \u2206 PQR \u2245 \u2206 QPS (by SAS congruence)<br \/>\n\u21d2 PR = QS.<br \/>\nHence, the diagonals of a rectangle are equal.<br \/>\nFrom \u2206 OPQ and \u2206 ORS,<br \/>\n\u2220OPQ = \u2220ORS (alternate angles)<br \/>\n\u2220OQP = \u2220OSR (alternate angles)<br \/>\nPQ = RS (opposite sides of a rectangle)<br \/>\nTherefore, \u2206OPQ \u2245 \u2206 ORS. (by ASA congruence)<br \/>\n\u21d2 OP = OR and OQ = OS.<br \/>\nThis shows that the diagonals of a rectangle bisect each other.<\/p>\n<p>Hence, the diagonals of a rectangle are equal and bisect each other.<\/p>\n<h3>Rhombus<\/h3>\n<p>The rhombus is a quadrilateral that consists of four sides with equal lengths.<\/p>\n<h4>Properties of Rhombus<\/h4>\n<ul>\n<li>The Rhombus consists of parallel and equal opposite sides. As it consists of parallel and equal opposite sides, it is said to be a parallelogram.<\/li>\n<li>All available sides (4 sides) are equal.<\/li>\n<li>Also, opposite angles in a rhombus are equal.<\/li>\n<li>Diagonals bisect each other.<\/li>\n<li>Diagonals of a rhombus intersect each other at right angles.<\/li>\n<li>Furthermore, Diagonals bisect opposite vertex angles.<\/li>\n<li>Every diagonal divides the rhombus into 2 congruent triangles.<\/li>\n<\/ul>\n<h4>Rhombus Formula<\/h4>\n<p>If b is the side, a and b are the two diagonals of the rhombus, then<br \/>\nArea = ab\/2 Square units.<br \/>\nPerimeter = 4b units<\/p>\n<h4>Diagonal Properties of a Rhombus<\/h4>\n<p>Prove that the diagonals of a rhombus bisect each other at right angles.<\/p>\n<p>Proof:<br \/>\nLet PQRS be a rhombus whose diagonals AC and BD intersect at point O.<br \/>\n<img loading=\"lazy\" class=\"alignnone wp-image-7282\" src=\"https:\/\/www.worksheetsbuddy.com\/wp-content\/uploads\/2021\/01\/Rhombus.jpg\" alt=\"Rhombus\" width=\"175\" height=\"120\" \/><\/p>\n<p>The diagonals of a parallelogram bisect each other. Also, we know that every rhombus is a parallelogram.<br \/>\nSo, the diagonals of a rhombus bisect each other.<br \/>\nTherefore, OP = OR and OQ = OS<br \/>\nFrom \u2206 ROQ and \u2206 ROS,<br \/>\nRQ = RS (sides of a rhombus)<br \/>\nRO = RO (common).<br \/>\nOQ = OS (proved)<br \/>\nTherefore, \u2206 ROQ \u2245 \u2206 ROS (by SSS congruence)<br \/>\n\u21d2 \u2220ROQ = \u2220ROS<br \/>\nBut, \u2220ROQ + \u2220ROS = 2 right angles (linear pair)<br \/>\nTherefore, \u2220ROQ = \u2220ROS = 1 right angle.<\/p>\n<p>Hence, the diagonals of a rhombus bisect each other at right angles.<\/p>\n<h3>Square<\/h3>\n<p>A square is a rectangle that has all equal sides.<\/p>\n<h4>Properties of Square<\/h4>\n<ul>\n<li>The opposite sides of a square are parallel.<\/li>\n<li>All 4 sides are equal in length.<\/li>\n<li>Diagonals are equal in length.<\/li>\n<li>Diagonals bisect opposite vertex angles.<\/li>\n<li>The interior angles of a square measure 90 degrees each.<\/li>\n<li>Diagonals bisect each other at right angles.<\/li>\n<li>It has 4 lines of symmetry \u2013 a horizontal, a vertical, and 2 diagonals.<\/li>\n<li>Each diagonal bisects the square into 2 congruent triangles.<\/li>\n<\/ul>\n<h4>Square Formula<\/h4>\n<p>If b is the side of the square, then<br \/>\nArea = b\u00b2 square units<br \/>\nPerimeter = 4b units.<\/p>\n<h4>Diagonal Properties of a Square<\/h4>\n<p>Prove that the diagonals of a square are equal and bisect each other at right angles.<\/p>\n<p>Proof:<br \/>\nWe know that the diagonals of a rectangle are equal.<br \/>\nAlso, every square is a rectangle.<br \/>\nTherefore, the diagonals of a square are equal.<br \/>\nAgain, the diagonals of a rhombus bisect each other at right angles. But, every square is a rhombus.<br \/>\nSo, the diagonals of a square bisect each other at right angles.<\/p>\n<p>Hence, the diagonals of a square are equal and also bisect each other at right angles.<\/p>\n<p>Note 1: If the diagonals of a quadrilateral are equal but it is not necessary to be a rectangle.<br \/>\nNote 2: If the diagonals of a quadrilateral interest at a point with right angles then also it is not necessary to become a rhombus.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Properties of a Rectangle Rhombus and Square is always a confusing concept for students. Learning every individual topic is important to score good marks in the exam. So, we have explained every individual topic clearly in a detailed manner in this article. Therefore, those who wish to learn the concepts of Parallelogram and its properties, &#8230; <a title=\"Properties of a Rectangle Rhombus and Square | Special Parallelograms Properties\" class=\"read-more\" href=\"https:\/\/bigideasmathanswers.com\/properties-of-a-rectangle-rhombus-and-square\/\" aria-label=\"More on Properties of a Rectangle Rhombus and Square | Special Parallelograms Properties\">Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[5],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/37019"}],"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=37019"}],"version-history":[{"count":1,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/37019\/revisions"}],"predecessor-version":[{"id":92612,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/37019\/revisions\/92612"}],"wp:attachment":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media?parent=37019"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/categories?post=37019"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/tags?post=37019"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}