{"id":11291,"date":"2022-07-01T01:17:32","date_gmt":"2022-06-30T19:47:32","guid":{"rendered":"https:\/\/mathemerize.com\/?p=11291"},"modified":"2022-07-01T01:18:02","modified_gmt":"2022-06-30T19:48:02","slug":"in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/","title":{"rendered":"In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR."},"content":{"rendered":"

Solution :<\/h2>\n

Given<\/strong> : O\u00a0 is any point within triangle PQR, AB || PQ and AC || PR\"similar<\/p>\n

To Prove<\/strong> :\u00a0 BC || QR<\/p>\n

Construction<\/strong> : Join BC<\/p>\n

Proof<\/strong> : In triangle OPQ,<\/p>\n

Given,\u00a0 \u00a0 \u00a0 AB || PQ<\/p>\n

By basic proportionality theorem,<\/p>\n

\\(OA\\over AP\\) = \\(OB\\over BQ\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 ………..(1)<\/p>\n

In triangle OPR,<\/p>\n

Given,\u00a0 \u00a0 \u00a0 \u00a0 AC || PR<\/p>\n

By basic proportionality theorem,<\/p>\n

\\(OA\\over AP\\) = \\(OC\\over CR\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0………..(2)<\/p>\n

From (1) and (2), we obtain that<\/p>\n

\\(OB\\over BQ\\) = \\(OC\\over CR\\)<\/p>\n

Hence, by converse of basic proportionality theorem, we have<\/p>\n

BC || QR<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

Solution : Given : O\u00a0 is any point within triangle PQR, AB || PQ and AC || PR To Prove :\u00a0 BC || QR Construction : Join BC Proof : In triangle OPQ, Given,\u00a0 \u00a0 \u00a0 AB || PQ By basic proportionality theorem, \\(OA\\over AP\\) = \\(OB\\over BQ\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 …<\/p>\n

In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.<\/span> Read More »<\/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":"\nIn the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR. - 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-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR. - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Solution : Given : O\u00a0 is any point within triangle PQR, AB || PQ and AC || PR To Prove :\u00a0 BC || QR Construction : Join BC Proof : In triangle OPQ, Given,\u00a0 \u00a0 \u00a0 AB || PQ By basic proportionality theorem, (OAover AP) = (OBover BQ)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 … In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR. Read More »\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2022-06-30T19:47:32+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-06-30T19:48:02+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-2-6-300x217.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-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.\",\"datePublished\":\"2022-06-30T19:47:32+00:00\",\"dateModified\":\"2022-06-30T19:48:02+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/\"},\"wordCount\":102,\"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-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/\",\"url\":\"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/\",\"name\":\"In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR. - Mathemerize\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2022-06-30T19:47:32+00:00\",\"dateModified\":\"2022-06-30T19:48:02+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.\"}]},{\"@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, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR. - 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-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/","og_locale":"en_US","og_type":"article","og_title":"In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR. - Mathemerize","og_description":"Solution : Given : O\u00a0 is any point within triangle PQR, AB || PQ and AC || PR To Prove :\u00a0 BC || QR Construction : Join BC Proof : In triangle OPQ, Given,\u00a0 \u00a0 \u00a0 AB || PQ By basic proportionality theorem, (OAover AP) = (OBover BQ)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 … In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR. Read More »","og_url":"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/","og_site_name":"Mathemerize","article_published_time":"2022-06-30T19:47:32+00:00","article_modified_time":"2022-06-30T19:48:02+00:00","og_image":[{"url":"https:\/\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-2-6-300x217.jpeg"}],"author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.","datePublished":"2022-06-30T19:47:32+00:00","dateModified":"2022-06-30T19:48:02+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/"},"wordCount":102,"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-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/","url":"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/","name":"In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR. - Mathemerize","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2022-06-30T19:47:32+00:00","dateModified":"2022-06-30T19:48:02+00:00","breadcrumb":{"@id":"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/in-the-figure-a-b-and-c-are-points-on-op-oq-and-or-respectively-such-that-ab-pq-and-ac-pr-show-that-bc-qr\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR."}]},{"@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\/"}]}},"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/11291"}],"collection":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/comments?post=11291"}],"version-history":[{"count":1,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/11291\/revisions"}],"predecessor-version":[{"id":11293,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/11291\/revisions\/11293"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=11291"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=11291"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=11291"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}