{"id":220496,"date":"2025-05-28T11:38:09","date_gmt":"2025-05-28T11:38:09","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=220496"},"modified":"2025-05-28T11:38:11","modified_gmt":"2025-05-28T11:38:11","slug":"are-the-triangles-below-right-triangles","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/28\/are-the-triangles-below-right-triangles\/","title":{"rendered":"Are The Triangles Below Right Triangles"},"content":{"rendered":"\n

Are The Triangles Below Right Triangles? If Not, What Kind Of Triangles Are They? 4195<\/p>\n\n\n\n

    \n
  1. Determine Whether The Numbers Can Represent The Side Lengths Of A Triangle. If They Can, Classify The Triangle As Acute, Right, Or Obtuse. Show Work!<\/li>\n\n\n\n
  2. 26, 35, 62<\/li>\n\n\n\n
  3. 29, 18, 14<\/li>\n\n\n\n
  4. 30, 78, 72<\/li>\n\n\n\n
  5. 17, 19, 22<\/li>\n<\/ol>\n\n\n\n
    \"\"<\/figure>\n\n\n\n

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

    Let’s evaluate each triangle using the Pythagorean Theorem<\/strong> and triangle inequality<\/strong>.<\/p>\n\n\n\n

    \n\n\n\n

    Part 1: Are the triangles right triangles?<\/h3>\n\n\n\n

    Problem 8: Sides = 9, 12, 15<\/h4>\n\n\n\n

    Check if 92+122=1529^2 + 12^2 = 15^2:
    81+144=22581 + 144 = 225
    225=225225 = 225 \u2705
    Answer<\/strong>: Right triangle<\/p>\n\n\n\n

    \n\n\n\n

    Problem 9: Sides = 15, 36, 4954\\sqrt{95}<\/h4>\n\n\n\n

    Calculate 495\u22484\u00d79.75\u2248394\\sqrt{95} \u2248 4 \u00d7 9.75 \u2248 39. Check:
    Is 152+362=(495)215^2 + 36^2 = (4\\sqrt{95})^2?
    Left: 225+1296=1521225 + 1296 = 1521
    Right: (495)2=16\u00d795=1520(4\\sqrt{95})^2 = 16 \u00d7 95 = 1520
    1521\u226015201521 \u2260 1520 \u274c
    Since hypotenuse\u00b2 < sum of squares of legs \u21d2 Acute triangle<\/strong><\/p>\n\n\n\n

    \n\n\n\n

    Problem 10: Sides = 7, 8, 113\\sqrt{113}<\/h4>\n\n\n\n

    113\u224810.63\\sqrt{113} \u2248 10.63
    Check if 72+82=(113)27^2 + 8^2 = (\\sqrt{113})^2:
    49+64=11349 + 64 = 113, and (113)2=113(\\sqrt{113})^2 = 113 \u2705
    Answer<\/strong>: Right triangle<\/p>\n\n\n\n

    \n\n\n\n

    Part 2: Can the numbers form a triangle? What kind?<\/h3>\n\n\n\n

    Use Triangle Inequality<\/strong>: The sum of the two shorter sides must be greater than the longest.<\/p>\n\n\n\n

    Problem 11: 26, 35, 62<\/h4>\n\n\n\n

    26 + 35 = 61 < 62 \u274c
    Answer<\/strong>: Not a triangle<\/p>\n\n\n\n

    \n\n\n\n

    Problem 12: 14, 18, 29<\/h4>\n\n\n\n

    14 + 18 = 32 > 29 \u2705
    14\u00b2 + 18\u00b2 = 196 + 324 = 520
    29\u00b2 = 841
    Since 520 < 841 \u21d2 Obtuse triangle<\/strong><\/p>\n\n\n\n

    \n\n\n\n

    Problem 13: 30, 72, 78<\/h4>\n\n\n\n

    30 + 72 = 102 > 78 \u2705
    30\u00b2 + 72\u00b2 = 900 + 5184 = 6084
    78\u00b2 = 6084
    6084 = 6084 \u2705
    Answer<\/strong>: Right triangle<\/p>\n\n\n\n

    \n\n\n\n

    Problem 14: 17, 19, 22<\/h4>\n\n\n\n

    17 + 19 = 36 > 22 \u2705
    17\u00b2 + 19\u00b2 = 289 + 361 = 650
    22\u00b2 = 484
    650 > 484 \u21d2 Acute triangle<\/strong><\/p>\n\n\n\n

    \n\n\n\n

    Summary:<\/h3>\n\n\n\n
    Problem<\/th>Triangle Type<\/th><\/tr><\/thead>
    8<\/td>Right<\/td><\/tr>
    9<\/td>Acute<\/td><\/tr>
    10<\/td>Right<\/td><\/tr>
    11<\/td>Not a triangle<\/td><\/tr>
    12<\/td>Obtuse<\/td><\/tr>
    13<\/td>Right<\/td><\/tr>
    14<\/td>Acute<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
    \n\n\n\n

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

    To determine the type of triangle, we apply two main methods: the Pythagorean Theorem<\/strong> and the Triangle Inequality Theorem<\/strong>. The Pythagorean Theorem states that for a triangle to be right-angled, the square of the longest side (hypotenuse) must equal the sum of the squares of the other two sides. If it\u2019s greater, the triangle is obtuse<\/strong>; if it\u2019s less, the triangle is acute<\/strong>.<\/p>\n\n\n\n

    In Problems 8 and 10, the side lengths satisfy the Pythagorean Theorem exactly, confirming that these are right triangles. For Problem 9, the calculated hypotenuse is slightly less than the sum of squares of the legs, meaning it is an acute triangle<\/strong>, as all angles are less than 90\u00b0.<\/p>\n\n\n\n

    For Problems 11\u201314, we first verify if the side lengths can form a triangle using the Triangle Inequality Theorem<\/strong>, which states the sum of any two sides must be greater than the third side. In Problem 11, this is not true, so the set cannot form a triangle.<\/p>\n\n\n\n

    For Problems 12 to 14, we then check the relationship between the square of the longest side and the sum of squares of the two shorter sides. If the longest side squared is greater, it\u2019s obtuse<\/strong> (like in Problem 12). If it\u2019s equal, the triangle is right<\/strong> (Problem 13). If it\u2019s less, it\u2019s acute<\/strong> (Problem 14).<\/p>\n\n\n\n

    These concepts help classify triangles by their angles using only side lengths, which is crucial in geometry and trigonometry.<\/p>\n\n\n\n

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

    Are The Triangles Below Right Triangles? If Not, What Kind Of Triangles Are They? 4195 The Correct Answer and Explanation is: Let’s evaluate each triangle using the Pythagorean Theorem and triangle inequality. Part 1: Are the triangles right triangles? Problem 8: Sides = 9, 12, 15 Check if 92+122=1529^2 + 12^2 = 15^2:81+144=22581 + 144 […]<\/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-220496","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/220496","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=220496"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/220496\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=220496"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=220496"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=220496"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}