{"id":20172,"date":"2025-06-14T08:10:44","date_gmt":"2025-06-14T08:10:44","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=20172"},"modified":"2025-06-14T08:10:47","modified_gmt":"2025-06-14T08:10:47","slug":"a-triangle-has-sides-with-lengths-7-24-and-25-1","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/a-triangle-has-sides-with-lengths-7-24-and-25-1\/","title":{"rendered":"A triangle has sides with lengths 7, 24, and 25. 1"},"content":{"rendered":"\n

A triangle has sides with lengths 7, 24, and 25. 1. Verify this is a Pythagorean triple. 2. Approximate the acute angles in this triangle.<\/p>\n\n\n\n

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

Let’s analyze the triangle with side lengths 7, 24, and 25.<\/p>\n\n\n\n

\n\n\n\n

1. Verify if this is a Pythagorean triple<\/h3>\n\n\n\n

A Pythagorean triple<\/strong> is a set of three positive integers a,b,ca, b, c such that: a2+b2=c2a^2 + b^2 = c^2<\/p>\n\n\n\n

where cc is the longest side (hypotenuse).<\/p>\n\n\n\n

Given the sides: 7, 24, and 25, the longest side is 25. Let’s check: 72+242=?7^2 + 24^2 = ?<\/p>\n\n\n\n

Calculate each: 72=497^2 = 49 242=57624^2 = 576<\/p>\n\n\n\n

Sum: 49+576=62549 + 576 = 625<\/p>\n\n\n\n

Now calculate 25225^2: 252=62525^2 = 625<\/p>\n\n\n\n

Since 72+242=2527^2 + 24^2 = 25^2<\/p>\n\n\n\n

this confirms that (7, 24, 25) is a Pythagorean triple<\/strong>, meaning this triangle is a right triangle.<\/p>\n\n\n\n

\n\n\n\n

2. Approximate the acute angles in this triangle<\/h3>\n\n\n\n

Since it is a right triangle, one angle is 90\u221890^\\circ. The other two angles are acute and sum to 90\u221890^\\circ.<\/p>\n\n\n\n

Let’s find the angles opposite the sides 7 and 24 using trigonometric functions. Let:<\/p>\n\n\n\n

    \n
  • \u03b8\\theta be the angle opposite the side of length 7,<\/li>\n\n\n\n
  • \u03d5\\phi be the angle opposite the side of length 24.<\/li>\n<\/ul>\n\n\n\n

    Using sine or tangent: sin\u2061\u03b8=oppositehypotenuse=725=0.28\\sin \\theta = \\frac{\\text{opposite}}{\\text{hypotenuse}} = \\frac{7}{25} = 0.28 \u03b8=sin\u2061\u22121(0.28)\u224816.26\u2218\\theta = \\sin^{-1}(0.28) \\approx 16.26^\\circ<\/p>\n\n\n\n

    For the other acute angle: \u03d5=90\u2218\u2212\u03b8\u224890\u2218\u221216.26\u2218=73.74\u2218\\phi = 90^\\circ – \\theta \\approx 90^\\circ – 16.26^\\circ = 73.74^\\circ<\/p>\n\n\n\n

    Or using tangent: tan\u2061\u03b8=724\u22480.2917\\tan \\theta = \\frac{7}{24} \\approx 0.2917 \u03b8=tan\u2061\u22121(0.2917)\u224816.26\u2218\\theta = \\tan^{-1}(0.2917) \\approx 16.26^\\circ<\/p>\n\n\n\n

    Both methods agree.<\/p>\n\n\n\n

    \n\n\n\n

    Summary<\/h3>\n\n\n\n
      \n
    • The triple (7, 24, 25) satisfies the Pythagorean theorem, confirming it is a right triangle.<\/li>\n\n\n\n
    • The acute angles are approximately 16.3\u221816.3^\\circ and 73.7\u221873.7^\\circ.<\/li>\n<\/ul>\n\n\n\n
      \n\n\n\n

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

      The Pythagorean theorem states that in a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side. By calculating 72+2427^2 + 24^2 and confirming it equals 25225^2, we prove the triangle is right-angled.<\/p>\n\n\n\n

      Knowing one angle is 90\u221890^\\circ, the other two angles must add to 90\u221890^\\circ. Using trigonometric functions such as sine or tangent with the known sides allows us to find the precise measures of these angles. Calculations show one angle is about 16.3\u00b0, making the other 73.7\u00b0. This knowledge helps in many practical fields such as construction, navigation, and physics, where accurate angle measurements are essential.<\/p>\n","protected":false},"excerpt":{"rendered":"

      A triangle has sides with lengths 7, 24, and 25. 1. Verify this is a Pythagorean triple. 2. Approximate the acute angles in this triangle. The correct answer and explanation is: Let’s analyze the triangle with side lengths 7, 24, and 25. 1. Verify if this is a Pythagorean triple A Pythagorean triple is a […]<\/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-20172","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/20172","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=20172"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/20172\/revisions"}],"predecessor-version":[{"id":20173,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/20172\/revisions\/20173"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=20172"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=20172"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=20172"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}