{"id":22942,"date":"2025-06-17T08:24:54","date_gmt":"2025-06-17T08:24:54","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=22942"},"modified":"2025-06-17T08:24:57","modified_gmt":"2025-06-17T08:24:57","slug":"a-right-triangle-has-sides-of-approximate-length-3-13-10-23-and-10-7","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/a-right-triangle-has-sides-of-approximate-length-3-13-10-23-and-10-7\/","title":{"rendered":"A right triangle has sides of approximate length 3.13, 10.23, and 10.7"},"content":{"rendered":"\n

A right triangle has sides of approximate length 3.13, 10.23, and 10.7. What is the angle opposite longest leg? Answer in degrees.<\/p>\n\n\n\n

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

To find the angle opposite the longest leg<\/strong> (which is 10.7 units<\/strong>), we must first identify whether the triangle is valid<\/strong> and then use trigonometry<\/strong> to solve the problem.<\/p>\n\n\n\n

\n\n\n\n

Step 1: Identify the Triangle Type<\/h3>\n\n\n\n

Given side lengths:<\/p>\n\n\n\n

    \n
  • a = 3.13<\/li>\n\n\n\n
  • b = 10.23<\/li>\n\n\n\n
  • c = 10.7<\/li>\n<\/ul>\n\n\n\n

    We are told it is a right triangle<\/strong>. In a right triangle, the longest side is the hypotenuse<\/strong>.
    Let\u2019s test if this triangle satisfies the Pythagorean Theorem<\/strong>: a2+b2=c2(check if it\u2019s a right triangle)a^2 + b^2 = c^2 \\quad \\text{(check if it’s a right triangle)} 3.132+10.232=9.7969+104.6529\u2248114.453.13^2 + 10.23^2 = 9.7969 + 104.6529 \u2248 114.45 10.72=114.4910.7^2 = 114.49<\/p>\n\n\n\n

    The values are very close, so we can confirm<\/strong> it’s a right triangle<\/strong> with sides 3.13, 10.23<\/strong>, and 10.7<\/strong> (hypotenuse).<\/p>\n\n\n\n

    \n\n\n\n

    Step 2: Use Sine Function<\/h3>\n\n\n\n

    To find the angle opposite the longest leg (10.7)<\/strong> \u2014 note that 10.7 is the hypotenuse<\/strong>, so we can’t find an angle opposite to it in a right triangle (the angle opposite the hypotenuse is always 90\u00b0<\/strong>).<\/p>\n\n\n\n

    However, the question seems to ask for the angle opposite the longest leg (not hypotenuse)<\/strong>. But 10.7 is not a leg<\/strong>, it\u2019s the hypotenuse<\/strong>. The longest leg<\/strong> here is 10.23<\/strong>, and we\u2019ll find the angle opposite that<\/strong>.<\/p>\n\n\n\n

    Let\u2019s denote:<\/p>\n\n\n\n

      \n
    • Opposite side = 10.23<\/li>\n\n\n\n
    • Hypotenuse = 10.7<\/li>\n<\/ul>\n\n\n\n

      Use the sine function: sin\u2061(\u03b8)=oppositehypotenuse=10.2310.7\u22480.9551\\sin(\\theta) = \\frac{\\text{opposite}}{\\text{hypotenuse}} = \\frac{10.23}{10.7} \u2248 0.9551<\/p>\n\n\n\n

      Now take the inverse sine: \u03b8=sin\u2061\u22121(0.9551)\u224872.83\u2218\\theta = \\sin^{-1}(0.9551) \u2248 72.83^\\circ<\/p>\n\n\n\n

      \n\n\n\n

      \u2705 Final Answer: Approximately 72.83\u00b0<\/strong><\/h3>\n\n\n\n
      \n\n\n\n

      \ud83d\udd0d Explanation (Summary):<\/h3>\n\n\n\n

      In a right triangle, the angle opposite a side can be found using trigonometric functions. Since the side 10.23 is the longest leg<\/strong> (not the hypotenuse), and the hypotenuse is 10.7, we used the sine function<\/strong>: sin\u2061(\u03b8)=opposite leghypotenuse\\sin(\\theta) = \\frac{\\text{opposite leg}}{\\text{hypotenuse}}<\/p>\n\n\n\n

      By solving using inverse sine, we found the angle is approximately 72.83 degrees<\/strong>, which is a reasonable value for a right triangle where one angle is already 90\u00b0.<\/p>\n","protected":false},"excerpt":{"rendered":"

      A right triangle has sides of approximate length 3.13, 10.23, and 10.7. What is the angle opposite longest leg? Answer in degrees. The correct answer and explanation is: To find the angle opposite the longest leg (which is 10.7 units), we must first identify whether the triangle is valid and then use trigonometry to solve […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-22942","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22942","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=22942"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22942\/revisions"}],"predecessor-version":[{"id":22943,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/22942\/revisions\/22943"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=22942"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=22942"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=22942"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}