{"id":143468,"date":"2022-07-17T00:12:37","date_gmt":"2022-07-16T18:42:37","guid":{"rendered":"https:\/\/ccssanswers.com\/?p=143468"},"modified":"2022-06-30T14:54:28","modified_gmt":"2022-06-30T09:24:28","slug":"converse-of-basic-proportionality-theorem","status":"publish","type":"post","link":"https:\/\/ccssanswers.com\/converse-of-basic-proportionality-theorem\/","title":{"rendered":"State &amp; Prove Converse of Basic Proportionality Theorem | Solved Problems on Converse of BPT"},"content":{"rendered":"<p>The basic proportionality theorem or Thales theorem was introduced by a mathematician named Thales. The converse of Thales theorem is also true. Students who have already learnt about the <a href=\"https:\/\/ccssanswers.com\/basic-proportionality-theorem\/\">Basic Proportionality Theorem<\/a> related to similarity can check the statement and proof of the converse of BPT. Get the complete details about the converse of basic proportionality theorem in the following sections.<\/p>\n<h2>Statement of Converse of Basic Proportionality Theorem<\/h2>\n<p>According to the converse of basic proportionality theorem, if a line segment is drawn to cut two sides of the triangle in equal ratio, then it is parallel to the third side.<\/p>\n<p>For example, in \u25b3ABC the line DE cuts the sides AB, AC in equal proportion. Such that DE is parallel to BC.<\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-120944\" src=\"https:\/\/eurekamathanswerkeys.com\/wp-content\/uploads\/2021\/10\/basic-proportionality-theorem-1.png\" alt=\"Basic Proportionality Theorem 1\" width=\"245\" height=\"269\" \/><\/p>\n<p><strong>Also, Check<\/strong><\/p>\n<ul>\n<li><a href=\"https:\/\/ccssanswers.com\/similar-triangles\/\">Similar Triangles<\/a><\/li>\n<li><a href=\"https:\/\/ccssanswers.com\/criteria-of-similarity-between-triangles\/\">Criteria of Similarity between Triangles<\/a><\/li>\n<li><a href=\"https:\/\/ccssanswers.com\/aa-criterion-of-similarity\/\">AA Criterion of Similarity<\/a><\/li>\n<\/ul>\n<h3>Converse of Basic Proportionality Theorem Proof<\/h3>\n<p>The following is the proof for the statement of the converse of basic proportionality theorem.<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-120985\" src=\"https:\/\/eurekamathanswerkeys.com\/wp-content\/uploads\/2021\/10\/converse-of-basic-proportionality-theorem-3.png\" alt=\"Converse of Basic Proportionality Theorem 3\" width=\"253\" height=\"247\" \/><\/p>\n<p><strong>Given: <\/strong>In \u25b3ABC, the line DE cuts the sides AB, AC in equal proportion. \\(\\frac { AD }{ BD } \\) = \\(\\frac { AE }{ CE } \\)<\/p>\n<p><strong>Construction:<\/strong> In triangle ABC, draw a line that meets AB at point D, AC at point E. Also, draw a line DF.<\/p>\n<p><strong>Proof:<\/strong><\/p>\n<p>Consider DE is not parallel to the side BC. Hence let us draw another line DF which is parallel to BC.<\/p>\n<p>By applying the basic proportionality theorem, \\(\\frac { AE }{ CE } \\) = \\(\\frac { AF }{ CF } \\)<\/p>\n<p>Add 1 to both sides of the equation.<\/p>\n<p>\\(\\frac { AE }{ CE } \\) + 1 = \\(\\frac { AF }{ CF } \\) + 1<\/p>\n<p>\\(\\frac { AE + CE }{ CE } \\) = \\(\\frac { AF + CF }{ CF } \\)<\/p>\n<p>\\(\\frac { AC }{ CE } \\) = \\(\\frac { AC }{ CF } \\)<\/p>\n<p>Therefore, CE = CF<\/p>\n<p>From the above statement, the points E and F are the same points and they coincide.<\/p>\n<p>Hence the line DE is parallel to BC and it proves the converse of the basic proportionality theorem.<\/p>\n<h3>Problems Using Converse of BPT<\/h3>\n<p><strong>Problem 1:<\/strong><br \/>\nIn a triangle ABC, D is a point on the side AB and E is a point on the side AC. And the lengths AB = 15 inches, AC = 17 inches, AD = 8 inches, AE = 7 inches. Is the line DE parallel to BC?<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-120969\" src=\"https:\/\/eurekamathanswerkeys.com\/wp-content\/uploads\/2021\/10\/converse-of-basic-proportionality-theorem-1.png\" alt=\"Converse of Basic Proportionality Theorem 1\" width=\"310\" height=\"245\" \/><\/p>\n<p><strong>Solution:<\/strong><br \/>\nFor the traingle ABC we have AB = 15 inches, AC = 17 inches, AD = 8 inches, AE = 7 inches<br \/>\nBD = AB &#8211; AD = 15 &#8211; 8 = 7 inches<br \/>\nCE = AC &#8211; AE = 17 &#8211; 7 = 10 inches<br \/>\nSo, \\(\\frac { AD }{ BD } \\) = \\(\\frac { 7 }{ 7 } \\) = 1 and \\(\\frac { AE }{ CE } \\) = 10\/10 = 1. That means \\(\\frac { AD }{ BD } \\) = \\(\\frac { AE }{ CE } \\). Applying the converse of basic proportionality theorem, the line segment DE is parallel to the side BC of the triangle ABC.<br \/>\n\u2234 DE is parallel to BC<\/p>\n<p><strong>Problem 2:<\/strong><br \/>\nP and Q are points on sides AB and AC respectively of \u25b3 ABC. For each of the following cases, state whether PQ \u2225 BC. AP = 4cm, PB = 4.5 cm, AQ = 4 cm, QC = 5 cm.<\/p>\n<p><strong>Solution:<\/strong><br \/>\nGiven that,<br \/>\nAP = 4cm, PB = 4.5 cm, AQ = 4 cm, QC = 5 cm<br \/>\nTo find PQ \u2225 BC<br \/>\n\\(\\frac { AP }{ PB } \\) = \\(\\frac { 4 }{ 4.5 } \\) = \\(\\frac { 8 }{ 9 } \\)<br \/>\n\\(\\frac { AQ }{ QC } \\) = \\(\\frac { 4 }{ 5 } \\)<br \/>\n\\(\\frac { AP }{ PB } \\) \u2260 \\(\\frac { AQ }{ QC } \\)<br \/>\nBasic proportionality theorem states that if a line is drawn parallel to the one side of the triangle the other sides in various points, then the other two sides are divided in the same ratio.<br \/>\nPQ is not parallel to BC.<\/p>\n<p><strong>Problem 3:<\/strong><br \/>\nABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that \\(\\frac { AO }{ BO } \\) = \\(\\frac { OC }{ OD } \\).<\/p>\n<p>Solution:<br \/>\n<img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-120982\" src=\"https:\/\/eurekamathanswerkeys.com\/wp-content\/uploads\/2021\/10\/converse-of-basic-proportionality-theorem-2.png\" alt=\"Converse of Basic Proportionality Theorem 2\" width=\"465\" height=\"278\" \/><br \/>\nDraw OM parallel to AB meeting BC at point M<br \/>\nIn \u25b3ACB,<br \/>\nIt is given that, OM is parallel to AB<br \/>\n\\(\\frac { OC }{ OA } \\) = \\(\\frac { CM }{ MB } \\) &#8212; (i)<br \/>\nIn \u25b3BDC<br \/>\nOM is parallel to CD<br \/>\n\\(\\frac { BM }{ MC } \\) = \\(\\frac { OB }{ OD } \\)<br \/>\nBy taking the reciprocals on both sides<br \/>\n\\(\\frac { CM }{ MB } \\) = \\(\\frac { OD }{ OB } \\) &#8212; (ii)<br \/>\n\\(\\frac { OC }{ OA } \\) = \\(\\frac { OD }{ OB } \\)<br \/>\n\\(\\frac { OC }{ OD } \\) = \\(\\frac { OA }{ OB } \\)<br \/>\nHence proved.<\/p>\n<h3>Frequently Asked Question&#8217;s<\/h3>\n<p><strong>1. What is the meaning of converse of BPT?<\/strong><\/p>\n<p>The converse of Thales theorem is if a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.<\/p>\n<p><strong>2. What is the difference between the basic proportion theorem and its converse?<\/strong><\/p>\n<p>Basic proportionality theorem is a line drawn parallel to one side of a triangle and cuts the other two sides in the same proportion. The converse of BPT is a line drawn to cut two sides of a triangle in the same ratio is parallel to the third side.<\/p>\n<p><strong>3. What is the BPT theorem?<\/strong><\/p>\n<p>Thales theorem states that a line drawn parallel to one side of a triangle and cutting the other two sides, divides the other two sides in equal proportion.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The basic proportionality theorem or Thales theorem was introduced by a mathematician named Thales. The converse of Thales theorem is also true. Students who have already learnt about the Basic Proportionality Theorem related to similarity can check the statement and proof of the converse of BPT. Get the complete details about the converse of basic &#8230; <a title=\"State &amp; Prove Converse of Basic Proportionality Theorem | Solved Problems on Converse of BPT\" class=\"read-more\" href=\"https:\/\/ccssanswers.com\/converse-of-basic-proportionality-theorem\/\" aria-label=\"More on State &amp; Prove Converse of Basic Proportionality Theorem | Solved Problems on Converse of BPT\">Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/143468"}],"collection":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/comments?post=143468"}],"version-history":[{"count":2,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/143468\/revisions"}],"predecessor-version":[{"id":143470,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/143468\/revisions\/143470"}],"wp:attachment":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/media?parent=143468"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/categories?post=143468"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/tags?post=143468"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}