{"id":237718,"date":"2025-06-17T13:41:35","date_gmt":"2025-06-17T13:41:35","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=237718"},"modified":"2025-06-17T13:41:37","modified_gmt":"2025-06-17T13:41:37","slug":"draw-an-inscribed-quadrilateral-abcdand-its-diagonals-intersecting-at-e","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/17\/draw-an-inscribed-quadrilateral-abcdand-its-diagonals-intersecting-at-e\/","title":{"rendered":"Draw an inscribed quadrilateral ABCDand its diagonals intersecting at E"},"content":{"rendered":"\n<p>Draw an inscribed quadrilateral ABCD<br>and its diagonals intersecting at E<br>. Name two pairs of similar triangles.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To draw an <strong>inscribed quadrilateral ABCD<\/strong> with diagonals intersecting at <strong>E<\/strong>, follow these steps and understand the geometric reasoning behind the pairs of similar triangles:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Drawing Instructions<\/strong><\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Draw a <strong>circle<\/strong>.<\/li>\n\n\n\n<li>Mark <strong>four points<\/strong> on the circle labeled <strong>A, B, C, and D<\/strong> in that order (moving clockwise or counterclockwise).<\/li>\n\n\n\n<li>Connect the points with straight lines to form the quadrilateral: draw segments <strong>AB, BC, CD, and DA<\/strong>.<\/li>\n\n\n\n<li>Draw the <strong>diagonals AC and BD<\/strong>. Let them intersect at point <strong>E<\/strong>.<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Identifying Similar Triangles<\/strong><\/h3>\n\n\n\n<p>When you draw diagonals <strong>AC and BD<\/strong>, they intersect at point <strong>E<\/strong>, forming <strong>four triangles<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u0394ABE<\/strong><\/li>\n\n\n\n<li><strong>\u0394CDE<\/strong><\/li>\n\n\n\n<li><strong>\u0394ADE<\/strong><\/li>\n\n\n\n<li><strong>\u0394CBE<\/strong><\/li>\n<\/ul>\n\n\n\n<p>Out of these, we can identify two <strong>pairs of similar triangles<\/strong>:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u0394ABE \u223c \u0394CDE<\/strong><\/li>\n\n\n\n<li><strong>\u0394ADE \u223c \u0394CBE<\/strong><\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Explanation<\/strong><\/h3>\n\n\n\n<p>A quadrilateral is said to be <strong>cyclic<\/strong> or <strong>inscribed<\/strong> when all its vertices lie on the same circle. This means opposite angles of the quadrilateral are <strong>supplementary<\/strong> (they add up to 180 degrees). Drawing the diagonals of a cyclic quadrilateral creates two pairs of vertical angles at their point of intersection, which is point E.<\/p>\n\n\n\n<p>Let\u2019s consider the triangles <strong>\u0394ABE<\/strong> and <strong>\u0394CDE<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Since both are inscribed in the circle and share the arc properties, <strong>angle ABE = angle CDE<\/strong> and <strong>angle BAE = angle DCE<\/strong> because they subtend the same arcs.<\/li>\n\n\n\n<li>The angle at <strong>E<\/strong> is <strong>vertically opposite<\/strong>, so <strong>angle AEB = angle CED<\/strong>.<\/li>\n\n\n\n<li>With two angles equal, <strong>\u0394ABE \u223c \u0394CDE<\/strong> by <strong>AA similarity criterion<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p>Now consider the second pair: <strong>\u0394ADE \u223c \u0394CBE<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Again, <strong>angle ADE = angle CBE<\/strong> and <strong>angle DAE = angle BCE<\/strong>, since they are inscribed angles subtending the same arcs.<\/li>\n\n\n\n<li>The angles at <strong>E<\/strong> are vertically opposite, hence <strong>angle DEA = angle CEB<\/strong>.<\/li>\n\n\n\n<li>Thus, these triangles also satisfy <strong>AA similarity<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p>This geometric relationship is a result of the properties of a circle and how angles subtended by the same arc or chord behave. Understanding these patterns helps identify similarities without needing measurements. These similar triangles are useful in proving other theorems and solving geometry problems involving proportions or angles.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-907.jpeg\" alt=\"\" class=\"wp-image-237719\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Draw an inscribed quadrilateral ABCDand its diagonals intersecting at E. Name two pairs of similar triangles. The Correct Answer and Explanation is: To draw an inscribed quadrilateral ABCD with diagonals intersecting at E, follow these steps and understand the geometric reasoning behind the pairs of similar triangles: Drawing Instructions Identifying Similar Triangles When you draw [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-237718","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/237718","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=237718"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/237718\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=237718"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=237718"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=237718"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}