{"id":11559,"date":"2022-07-18T14:52:25","date_gmt":"2022-07-18T09:22:25","guid":{"rendered":"https:\/\/mathemerize.com\/?p=11559"},"modified":"2022-07-18T14:52:29","modified_gmt":"2022-07-18T09:22:29","slug":"in-the-figure-two-chords-ab-and-cd-of-a-circle-intersect-each-other-at-the-point-p-when-produced-outside-the-circle-prove-that-i-triangle-pac-triangle-pdb-ii-pa-pb-pc","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/in-the-figure-two-chords-ab-and-cd-of-a-circle-intersect-each-other-at-the-point-p-when-produced-outside-the-circle-prove-that-i-triangle-pac-triangle-pdb-ii-pa-pb-pc\/","title":{"rendered":"In the figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that  (i)  \\(\\triangle\\) PAC ~ \\(\\triangle\\) PDB     (ii) PA.PB = PC.PD"},"content":{"rendered":"<h2>Solution :<\/h2>\n<p><strong>(i)<\/strong>\u00a0 In \\(\\triangle\\)s PAC and PDB, we have :<img decoding=\"async\" loading=\"lazy\" class=\" wp-image-11583 alignright\" src=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-6-8.jpeg?resize=197%2C157&#038;ssl=1\" alt=\"circle\" width=\"197\" height=\"157\" srcset=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-6-8.jpeg?resize=300%2C239&amp;ssl=1 300w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-6-8.jpeg?resize=1024%2C816&amp;ssl=1 1024w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-6-8.jpeg?resize=768%2C612&amp;ssl=1 768w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-6-8.jpeg?resize=1536%2C1224&amp;ssl=1 1536w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-6-8.jpeg?w=1700&amp;ssl=1 1700w\" sizes=\"(max-width: 197px) 100vw, 197px\" data-recalc-dims=\"1\" \/><\/p>\n<p>\\(\\angle\\) APC = \\(\\angle\\) DPB\u00a0 \u00a0 \u00a0 \u00a0 \u00a0(common)<\/p>\n<p>\\(\\angle\\) PAC = \\(\\angle\\) PDB<\/p>\n<p>[\\(\\therefore\\)\u00a0 \\(\\angle\\) BAC = 180 &#8211; \\(\\angle\\) PAC\u00a0 and \\(\\angle\\) PDB = \\(\\angle\\) CDB = 180 &#8211; (180 &#8211; \\(\\angle\\) PAC) = \\(\\angle\\) PAC]<\/p>\n<p>\\(\\therefore\\)\u00a0 By AA similarity, we have :<\/p>\n<p><strong>\\(\\triangle\\) PAC ~ \\(\\triangle\\) PDB<\/strong><\/p>\n<p><strong>(ii)<\/strong>\u00a0 Since \\(\\triangle\\)s PAC ~ PDB, therefore<\/p>\n<p>\\(PA\\over PD\\) = \\(PC\\over PB\\)\u00a0 or\u00a0 <strong>PA.PB = PC.PD.<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution : (i)\u00a0 In \\(\\triangle\\)s PAC and PDB, we have : \\(\\angle\\) APC = \\(\\angle\\) DPB\u00a0 \u00a0 \u00a0 \u00a0 \u00a0(common) \\(\\angle\\) PAC = \\(\\angle\\) PDB [\\(\\therefore\\)\u00a0 \\(\\angle\\) BAC = 180 &#8211; \\(\\angle\\) PAC\u00a0 and \\(\\angle\\) PDB = \\(\\angle\\) CDB = 180 &#8211; (180 &#8211; \\(\\angle\\) PAC) = \\(\\angle\\) PAC] \\(\\therefore\\)\u00a0 By AA similarity, we have &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/in-the-figure-two-chords-ab-and-cd-of-a-circle-intersect-each-other-at-the-point-p-when-produced-outside-the-circle-prove-that-i-triangle-pac-triangle-pdb-ii-pa-pb-pc\/\"> <span class=\"screen-reader-text\">In the figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that  (i)  \\(\\triangle\\) PAC ~ \\(\\triangle\\) PDB     (ii) PA.PB = PC.PD<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[43,912],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.12 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>In the figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that (i) \\(\\triangle\\) PAC ~ \\(\\triangle\\) PDB   (ii) PA.PB = PC.PD - Mathemerize<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/in-the-figure-two-chords-ab-and-cd-of-a-circle-intersect-each-other-at-the-point-p-when-produced-outside-the-circle-prove-that-i-triangle-pac-triangle-pdb-ii-pa-pb-pc\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"In the figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that (i) \\(\\triangle\\) PAC ~ \\(\\triangle\\) PDB   (ii) PA.PB = PC.PD - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Solution : (i)\u00a0 In (triangle)s PAC and PDB, we have : (angle) APC = (angle) DPB\u00a0 \u00a0 \u00a0 \u00a0 \u00a0(common) (angle) PAC = (angle) PDB [(therefore)\u00a0 (angle) BAC = 180 &#8211; (angle) PAC\u00a0 and (angle) PDB = (angle) CDB = 180 &#8211; (180 &#8211; (angle) PAC) = (angle) PAC] (therefore)\u00a0 By AA similarity, we have &hellip; In the figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that (i) (triangle) PAC ~ (triangle) PDB   (ii) PA.PB = PC.PD Read More &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/in-the-figure-two-chords-ab-and-cd-of-a-circle-intersect-each-other-at-the-point-p-when-produced-outside-the-circle-prove-that-i-triangle-pac-triangle-pdb-ii-pa-pb-pc\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2022-07-18T09:22:25+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-07-18T09:22:29+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-6-8-300x239.jpeg\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/in-the-figure-two-chords-ab-and-cd-of-a-circle-intersect-each-other-at-the-point-p-when-produced-outside-the-circle-prove-that-i-triangle-pac-triangle-pdb-ii-pa-pb-pc\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/in-the-figure-two-chords-ab-and-cd-of-a-circle-intersect-each-other-at-the-point-p-when-produced-outside-the-circle-prove-that-i-triangle-pac-triangle-pdb-ii-pa-pb-pc\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"In the figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that (i) \\\\(\\\\triangle\\\\) PAC ~ \\\\(\\\\triangle\\\\) PDB (ii) PA.PB = PC.PD\",\"datePublished\":\"2022-07-18T09:22:25+00:00\",\"dateModified\":\"2022-07-18T09:22:29+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/in-the-figure-two-chords-ab-and-cd-of-a-circle-intersect-each-other-at-the-point-p-when-produced-outside-the-circle-prove-that-i-triangle-pac-triangle-pdb-ii-pa-pb-pc\/\"},\"wordCount\":99,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"articleSection\":[\"Maths Questions\",\"Similar Triangles Questions\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/in-the-figure-two-chords-ab-and-cd-of-a-circle-intersect-each-other-at-the-point-p-when-produced-outside-the-circle-prove-that-i-triangle-pac-triangle-pdb-ii-pa-pb-pc\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/in-the-figure-two-chords-ab-and-cd-of-a-circle-intersect-each-other-at-the-point-p-when-produced-outside-the-circle-prove-that-i-triangle-pac-triangle-pdb-ii-pa-pb-pc\/\",\"url\":\"https:\/\/mathemerize.com\/in-the-figure-two-chords-ab-and-cd-of-a-circle-intersect-each-other-at-the-point-p-when-produced-outside-the-circle-prove-that-i-triangle-pac-triangle-pdb-ii-pa-pb-pc\/\",\"name\":\"In the figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. 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Prove that (i) \\\\(\\\\triangle\\\\) PAC ~ \\\\(\\\\triangle\\\\) PDB (ii) PA.PB = PC.PD\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - Study Math Online\",\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathemerize.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathemerize.com\/#organization\",\"name\":\"Mathemerize\",\"url\":\"https:\/\/mathemerize.com\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"contentUrl\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"width\":140,\"height\":96,\"caption\":\"Mathemerize\"},\"image\":{\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.instagram.com\/mathemerize\/\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\",\"name\":\"mathemerize\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"caption\":\"mathemerize\"},\"sameAs\":[\"https:\/\/mathemerize.com\"],\"url\":\"https:\/\/mathemerize.com\/author\/mathemerize\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"In the figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that (i) \\(\\triangle\\) PAC ~ \\(\\triangle\\) PDB   (ii) PA.PB = PC.PD - Mathemerize","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathemerize.com\/in-the-figure-two-chords-ab-and-cd-of-a-circle-intersect-each-other-at-the-point-p-when-produced-outside-the-circle-prove-that-i-triangle-pac-triangle-pdb-ii-pa-pb-pc\/","og_locale":"en_US","og_type":"article","og_title":"In the figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that (i) \\(\\triangle\\) PAC ~ \\(\\triangle\\) PDB   (ii) PA.PB = PC.PD - Mathemerize","og_description":"Solution : (i)\u00a0 In (triangle)s PAC and PDB, we have : (angle) APC = (angle) DPB\u00a0 \u00a0 \u00a0 \u00a0 \u00a0(common) (angle) PAC = (angle) PDB [(therefore)\u00a0 (angle) BAC = 180 &#8211; (angle) PAC\u00a0 and (angle) PDB = (angle) CDB = 180 &#8211; (180 &#8211; (angle) PAC) = (angle) PAC] (therefore)\u00a0 By AA similarity, we have &hellip; In the figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. 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