{"id":201404,"date":"2025-03-15T09:57:03","date_gmt":"2025-03-15T09:57:03","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=201404"},"modified":"2025-03-15T09:57:05","modified_gmt":"2025-03-15T09:57:05","slug":"the-sides-of-the-triangle-are-5-8-10-respectively","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/15\/the-sides-of-the-triangle-are-5-8-10-respectively\/","title":{"rendered":"The sides of the triangle are 5, 8, 10 respectively"},"content":{"rendered":"\n

The sides of the triangle are 5, 8, 10 respectively. Find the lengths of the medians.<\/p>\n\n\n\n

The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n

Given a triangle with sides 5, 8, and 10, we can calculate the lengths of the medians using Apollonius’s theorem<\/strong> or formula for the medians<\/strong>.<\/p>\n\n\n\n

Step 1: Understanding Medians<\/h3>\n\n\n\n

A median<\/strong> of a triangle is a line segment joining a vertex to the midpoint of the opposite side. The three medians of a triangle are typically denoted as ( m_a ), ( m_b ), and ( m_c ), corresponding to the sides ( a ), ( b ), and ( c ) respectively.<\/p>\n\n\n\n

The length of a median<\/strong> from a vertex opposite to side ( a ) is given by the formula:<\/p>\n\n\n\n

[
m_a = \\sqrt{\\frac{2b^2 + 2c^2 – a^2}{4}}
]<\/p>\n\n\n\n

where:<\/p>\n\n\n\n

    \n
  • ( a ), ( b ), and ( c ) are the lengths of the sides of the triangle.<\/li>\n<\/ul>\n\n\n\n

    In this case, the sides are 5, 8, and 10. So, we have ( a = 5 ), ( b = 8 ), and ( c = 10 ).<\/p>\n\n\n\n

    Step 2: Calculate the Medians<\/h3>\n\n\n\n

    Median ( m_a ) (opposite side 5):<\/h4>\n\n\n\n

    [
    m_a = \\sqrt{\\frac{2(8^2) + 2(10^2) – 5^2}{4}} = \\sqrt{\\frac{2(64) + 2(100) – 25}{4}} = \\sqrt{\\frac{128 + 200 – 25}{4}} = \\sqrt{\\frac{303}{4}} = \\sqrt{75.75} \\approx 8.69
    ]<\/p>\n\n\n\n

    Median ( m_b ) (opposite side 8):<\/h4>\n\n\n\n

    [
    m_b = \\sqrt{\\frac{2(5^2) + 2(10^2) – 8^2}{4}} = \\sqrt{\\frac{2(25) + 2(100) – 64}{4}} = \\sqrt{\\frac{50 + 200 – 64}{4}} = \\sqrt{\\frac{186}{4}} = \\sqrt{46.5} \\approx 6.82
    ]<\/p>\n\n\n\n

    Median ( m_c ) (opposite side 10):<\/h4>\n\n\n\n

    [
    m_c = \\sqrt{\\frac{2(5^2) + 2(8^2) – 10^2}{4}} = \\sqrt{\\frac{2(25) + 2(64) – 100}{4}} = \\sqrt{\\frac{50 + 128 – 100}{4}} = \\sqrt{\\frac{78}{4}} = \\sqrt{19.5} \\approx 4.41
    ]<\/p>\n\n\n\n

    Step 3: Final Results<\/h3>\n\n\n\n

    The lengths of the medians are approximately:<\/p>\n\n\n\n

      \n
    • Median ( m_a ) \u2248 8.69<\/li>\n\n\n\n
    • Median ( m_b ) \u2248 6.82<\/li>\n\n\n\n
    • Median ( m_c ) \u2248 4.41<\/li>\n<\/ul>\n\n\n\n

      Conclusion<\/h3>\n\n\n\n

      To summarize, for a triangle with sides 5, 8, and 10, the medians have the following lengths:<\/p>\n\n\n\n

        \n
      • ( m_a \\approx 8.69 )<\/li>\n\n\n\n
      • ( m_b \\approx 6.82 )<\/li>\n\n\n\n
      • ( m_c \\approx 4.41 )<\/li>\n<\/ul>\n\n\n\n

        These results were derived using the median length formula, which depends on the side lengths of the triangle.<\/p>\n","protected":false},"excerpt":{"rendered":"

        The sides of the triangle are 5, 8, 10 respectively. Find the lengths of the medians. The correct answer and explanation is : Given a triangle with sides 5, 8, and 10, we can calculate the lengths of the medians using Apollonius’s theorem or formula for the medians. Step 1: Understanding Medians A median of […]<\/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-201404","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/201404","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=201404"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/201404\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=201404"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=201404"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=201404"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}