{"id":11353,"date":"2022-07-07T23:00:33","date_gmt":"2022-07-07T17:30:33","guid":{"rendered":"https:\/\/mathemerize.com\/?p=11353"},"modified":"2022-07-07T23:05:40","modified_gmt":"2022-07-07T17:35:40","slug":"in-the-figure-altitudes-ad-and-ce-of-triangle-abc-intersect-each-other-at-the-point-p-show-that-i-triangle-aep-triangle-cdp-ii-triangle-abd-triangle","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/in-the-figure-altitudes-ad-and-ce-of-triangle-abc-intersect-each-other-at-the-point-p-show-that-i-triangle-aep-triangle-cdp-ii-triangle-abd-triangle\/","title":{"rendered":"In the figure, altitudes AD and CE of \\(\\triangle\\) ABC intersect each other at the point P. Show that : (i) \\(\\triangle\\) AEP ~ \\(\\triangle\\) CDP  (ii) \\(\\triangle\\) ABD ~ \\(\\triangle\\) CBE (iii) \\(\\triangle\\) AEP ~ \\(\\triangle\\) ADB (iv) \\(\\triangle\\) PDC ~ \\(\\triangle\\) BEC"},"content":{"rendered":"<h2>Solution :<\/h2>\n<p><strong>(i)<\/strong>\u00a0 In <img decoding=\"async\" loading=\"lazy\" class=\" wp-image-11355 alignright\" src=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-3-7.jpeg?resize=255%2C183&#038;ssl=1\" alt=\"triangle\" width=\"255\" height=\"183\" srcset=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-3-7.jpeg?resize=300%2C215&amp;ssl=1 300w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-3-7.jpeg?resize=1024%2C733&amp;ssl=1 1024w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-3-7.jpeg?resize=768%2C550&amp;ssl=1 768w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-3-7.jpeg?resize=1536%2C1100&amp;ssl=1 1536w, https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-3-7.jpeg?w=1700&amp;ssl=1 1700w\" sizes=\"(max-width: 255px) 100vw, 255px\" data-recalc-dims=\"1\" \/>\\(\\triangle\\) AEP and CDP, we have<\/p>\n<p>\\(\\angle\\) AEP = \\(\\angle\\) CDP = 90<\/p>\n<p>\\(\\angle\\) APE = \\(\\angle\\) CPD\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(vertically opposite angles)<\/p>\n<p>\\(\\therefore\\)\u00a0 By AA similarity, we have :<\/p>\n<p><strong>\\(\\triangle\\) AEP ~ \\(\\triangle\\) CDP<\/strong><\/p>\n<p><strong>(ii)<\/strong>\u00a0 In \\(\\triangle\\) ABD and CBE, we have<\/p>\n<p>\\(\\angle\\) ABD = \\(\\angle\\) CBE\u00a0 \u00a0 \u00a0 \u00a0 \u00a0(common angle)<\/p>\n<p>\\(\\angle\\) ADB = \\(\\angle\\) CEB = 90<\/p>\n<p>\\(\\therefore\\)\u00a0 By AA similarity, we have :<\/p>\n<p><strong>\\(\\triangle\\) ABD ~ \\(\\triangle\\) CBE<\/strong><\/p>\n<p><strong>(iii)<\/strong>\u00a0 In \\(\\triangle\\) AEP and ADB, we have<\/p>\n<p>\\(\\angle\\) AEP = \\(\\angle\\) ADB = 90<\/p>\n<p>\\(\\angle\\) PAE = \\(\\angle\\)\u00a0 DAB\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (common angles)<\/p>\n<p>\\(\\therefore\\)\u00a0 By AA similarity, we have :<\/p>\n<p><strong>\\(\\triangle\\) AEP ~ \\(\\triangle\\) ADB<\/strong><\/p>\n<p><strong>(iv)<\/strong>\u00a0 In \\(\\triangle\\) PDC and BEC, we have<\/p>\n<p>\\(\\angle\\) PDC = \\(\\angle\\) BEC = 90<\/p>\n<p>\\(\\angle\\) PCD = \\(\\angle\\) ECB\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(common angles)<\/p>\n<p>\\(\\therefore\\)\u00a0 By AA similarity, we have :<\/p>\n<p><strong>\\(\\triangle\\) PDC ~ \\(\\triangle\\) BEC<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution : (i)\u00a0 In \\(\\triangle\\) AEP and CDP, we have \\(\\angle\\) AEP = \\(\\angle\\) CDP = 90 \\(\\angle\\) APE = \\(\\angle\\) CPD\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(vertically opposite angles) \\(\\therefore\\)\u00a0 By AA similarity, we have : \\(\\triangle\\) AEP ~ \\(\\triangle\\) CDP (ii)\u00a0 In \\(\\triangle\\) ABD and CBE, we have \\(\\angle\\) ABD = \\(\\angle\\) CBE\u00a0 \u00a0 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/in-the-figure-altitudes-ad-and-ce-of-triangle-abc-intersect-each-other-at-the-point-p-show-that-i-triangle-aep-triangle-cdp-ii-triangle-abd-triangle\/\"> <span class=\"screen-reader-text\">In the figure, altitudes AD and CE of \\(\\triangle\\) ABC intersect each other at the point P. Show that : (i) \\(\\triangle\\) AEP ~ \\(\\triangle\\) CDP  (ii) \\(\\triangle\\) ABD ~ \\(\\triangle\\) CBE (iii) \\(\\triangle\\) AEP ~ \\(\\triangle\\) ADB (iv) \\(\\triangle\\) PDC ~ \\(\\triangle\\) BEC<\/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, altitudes AD and CE of \\(\\triangle\\) ABC intersect each other at the point P. Show that : (i) \\(\\triangle\\) AEP ~ \\(\\triangle\\) CDP (ii) \\(\\triangle\\) ABD ~ \\(\\triangle\\) CBE (iii) \\(\\triangle\\) AEP ~ \\(\\triangle\\) ADB (iv) \\(\\triangle\\) PDC ~ \\(\\triangle\\) BEC - 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-altitudes-ad-and-ce-of-triangle-abc-intersect-each-other-at-the-point-p-show-that-i-triangle-aep-triangle-cdp-ii-triangle-abd-triangle\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"In the figure, altitudes AD and CE of \\(\\triangle\\) ABC intersect each other at the point P. Show that : (i) \\(\\triangle\\) AEP ~ \\(\\triangle\\) CDP (ii) \\(\\triangle\\) ABD ~ \\(\\triangle\\) CBE (iii) \\(\\triangle\\) AEP ~ \\(\\triangle\\) ADB (iv) \\(\\triangle\\) PDC ~ \\(\\triangle\\) BEC - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Solution : (i)\u00a0 In (triangle) AEP and CDP, we have (angle) AEP = (angle) CDP = 90 (angle) APE = (angle) CPD\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(vertically opposite angles) (therefore)\u00a0 By AA similarity, we have : (triangle) AEP ~ (triangle) CDP (ii)\u00a0 In (triangle) ABD and CBE, we have (angle) ABD = (angle) CBE\u00a0 \u00a0 &hellip; In the figure, altitudes AD and CE of (triangle) ABC intersect each other at the point P. Show that : (i) (triangle) AEP ~ (triangle) CDP (ii) (triangle) ABD ~ (triangle) CBE (iii) (triangle) AEP ~ (triangle) ADB (iv) (triangle) PDC ~ (triangle) BEC Read More &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/in-the-figure-altitudes-ad-and-ce-of-triangle-abc-intersect-each-other-at-the-point-p-show-that-i-triangle-aep-triangle-cdp-ii-triangle-abd-triangle\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2022-07-07T17:30:33+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-07-07T17:35:40+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-3-7-300x215.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-altitudes-ad-and-ce-of-triangle-abc-intersect-each-other-at-the-point-p-show-that-i-triangle-aep-triangle-cdp-ii-triangle-abd-triangle\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/in-the-figure-altitudes-ad-and-ce-of-triangle-abc-intersect-each-other-at-the-point-p-show-that-i-triangle-aep-triangle-cdp-ii-triangle-abd-triangle\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"In the figure, altitudes AD and CE of \\\\(\\\\triangle\\\\) ABC intersect each other at the point P. 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Show that : (i) \\(\\triangle\\) AEP ~ \\(\\triangle\\) CDP (ii) \\(\\triangle\\) ABD ~ \\(\\triangle\\) CBE (iii) \\(\\triangle\\) AEP ~ \\(\\triangle\\) ADB (iv) \\(\\triangle\\) PDC ~ \\(\\triangle\\) BEC - 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-altitudes-ad-and-ce-of-triangle-abc-intersect-each-other-at-the-point-p-show-that-i-triangle-aep-triangle-cdp-ii-triangle-abd-triangle\/","og_locale":"en_US","og_type":"article","og_title":"In the figure, altitudes AD and CE of \\(\\triangle\\) ABC intersect each other at the point P. Show that : (i) \\(\\triangle\\) AEP ~ \\(\\triangle\\) CDP (ii) \\(\\triangle\\) ABD ~ \\(\\triangle\\) CBE (iii) \\(\\triangle\\) AEP ~ \\(\\triangle\\) ADB (iv) \\(\\triangle\\) PDC ~ \\(\\triangle\\) BEC - Mathemerize","og_description":"Solution : (i)\u00a0 In (triangle) AEP and CDP, we have (angle) AEP = (angle) CDP = 90 (angle) APE = (angle) CPD\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(vertically opposite angles) (therefore)\u00a0 By AA similarity, we have : (triangle) AEP ~ (triangle) CDP (ii)\u00a0 In (triangle) ABD and CBE, we have (angle) ABD = (angle) CBE\u00a0 \u00a0 &hellip; In the figure, altitudes AD and CE of (triangle) ABC intersect each other at the point P. Show that : (i) (triangle) AEP ~ (triangle) CDP (ii) (triangle) ABD ~ (triangle) CBE (iii) (triangle) AEP ~ (triangle) ADB (iv) (triangle) PDC ~ (triangle) BEC Read More &raquo;","og_url":"https:\/\/mathemerize.com\/in-the-figure-altitudes-ad-and-ce-of-triangle-abc-intersect-each-other-at-the-point-p-show-that-i-triangle-aep-triangle-cdp-ii-triangle-abd-triangle\/","og_site_name":"Mathemerize","article_published_time":"2022-07-07T17:30:33+00:00","article_modified_time":"2022-07-07T17:35:40+00:00","og_image":[{"url":"https:\/\/mathemerize.com\/wp-content\/uploads\/2022\/07\/similartriangles6-3-7-300x215.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-altitudes-ad-and-ce-of-triangle-abc-intersect-each-other-at-the-point-p-show-that-i-triangle-aep-triangle-cdp-ii-triangle-abd-triangle\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/in-the-figure-altitudes-ad-and-ce-of-triangle-abc-intersect-each-other-at-the-point-p-show-that-i-triangle-aep-triangle-cdp-ii-triangle-abd-triangle\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"In the figure, altitudes AD and CE of \\(\\triangle\\) ABC intersect each other at the point P. 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