{"id":228269,"date":"2025-06-07T07:27:15","date_gmt":"2025-06-07T07:27:15","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=228269"},"modified":"2025-06-07T07:27:17","modified_gmt":"2025-06-07T07:27:17","slug":"theorem-6-the-median-of-a-trapezoid-is-parallel-to-each-base-and-its-length-is-one-half-the-sum-of-the-lengths-of-the-bases","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/07\/theorem-6-the-median-of-a-trapezoid-is-parallel-to-each-base-and-its-length-is-one-half-the-sum-of-the-lengths-of-the-bases\/","title":{"rendered":"Theorem 6. The median of a trapezoid is parallel to each base and its length is one-half the sum of the lengths of the bases."},"content":{"rendered":"\n

Theorem 6. The median of a trapezoid is parallel to each base and its length is one-half the sum of the lengths of the bases. Example: Given: Trapezoid MINS with median
Prove:<\/p>\n\n\n\n

,<\/p>\n\n\n\n

,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Correct Answer:<\/strong><\/h3>\n\n\n\n

Given: Trapezoid MINSMINSMINS with median TR\u203e\\overline{TR}TR<\/p>\n\n\n\n

To Prove:<\/p>\n\n\n\n

    \n
  1. TR\u203e\u2225IN\u203e\\overline{TR} \\parallel \\overline{IN}TR\u2225IN<\/li>\n\n\n\n
  2. TR\u203e\u2225MS\u203e\\overline{TR} \\parallel \\overline{MS}TR\u2225MS<\/li>\n\n\n\n
  3. TP=12(MS\u203e+IN\u203e)TP = \\frac{1}{2} (\\overline{MS} + \\overline{IN})TP=21\u200b(MS+IN)<\/li>\n<\/ol>\n\n\n\n
    \n\n\n\n

    Explanation<\/strong><\/h3>\n\n\n\n

    Theorem 6 in geometry states that the median of a trapezoid is parallel to both bases<\/strong>, and its length is the average (one-half the sum) of the lengths of the two bases<\/strong>.<\/p>\n\n\n\n

    In the given figure, MINSMINSMINS is a trapezoid where MS\u203e\\overline{MS}MS and IN\u203e\\overline{IN}IN are the two bases<\/strong>, and TR\u203e\\overline{TR}TR is the median<\/strong>. The median connects the midpoints of the non-parallel sides MI\u203e\\overline{MI}MI and SN\u203e\\overline{SN}SN. This is evident from point TTT lying midway on MI\u203e\\overline{MI}MI and point RRR lying midway on SN\u203e\\overline{SN}SN.<\/p>\n\n\n\n

    By Theorem 6, the median TR\u203e\\overline{TR}TR must be parallel<\/strong> to both IN\u203e\\overline{IN}IN and MS\u203e\\overline{MS}MS. Thus, we have:<\/p>\n\n\n\n

      \n
    • TR\u203e\u2225IN\u203e\\overline{TR} \\parallel \\overline{IN}TR\u2225IN<\/li>\n\n\n\n
    • TR\u203e\u2225MS\u203e\\overline{TR} \\parallel \\overline{MS}TR\u2225MS<\/li>\n<\/ul>\n\n\n\n

      This parallelism arises from the properties of similar triangles and midsegment theorems. Since TTT and RRR are midpoints, the line segment connecting them (i.e., TR\u203e\\overline{TR}TR) must be parallel to both bases.<\/p>\n\n\n\n

      Furthermore, the length<\/strong> of the median TR\u203e\\overline{TR}TR is the average<\/strong> of the lengths of the two bases. This leads to the equation:TP=12(MS\u203e+IN\u203e)TP = \\frac{1}{2} (\\overline{MS} + \\overline{IN})TP=21\u200b(MS+IN)<\/p>\n\n\n\n

      This is a direct application of the trapezoid midsegment theorem<\/strong>, which ensures that the segment connecting the midpoints of the non-parallel sides of a trapezoid has a length equal to the average of the two bases.<\/p>\n\n\n\n

      In summary, the problem illustrates a fundamental property of trapezoids: the median is both parallel to the bases and has a length equal to the mean of the bases.<\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      Theorem 6. The median of a trapezoid is parallel to each base and its length is one-half the sum of the lengths of the bases. Example: Given: Trapezoid MINS with medianProve: , , The Correct Answer and Explanation is: Correct Answer: Given: Trapezoid MINSMINSMINS with median TR\u203e\\overline{TR}TR To Prove: Explanation Theorem 6 in geometry states […]<\/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-228269","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/228269","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=228269"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/228269\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=228269"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=228269"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=228269"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}