Multiple choice

If the perimeter of a right-angled triangle is 56 cm and the area of the triangle is 84 sq.cm, then the length of the hypotenuse is

  1. 25 cm

  2. 50 cm

  3. 7 cm

  4. 24 cm
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A Correct answer
Explanation

Perimeter = a + b + c = 56. Area = 0.5 * a * b = 84 => ab = 168. a^2 + b^2 = c^2. (a+b)^2 - 2ab = c^2. (56-c)^2 - 2(168) = c^2. 3136 - 112c + c^2 - 336 = c^2. 2800 = 112c. c = 25.

AI explanation

Let the legs be a and b, and the hypotenuse be c. The area formula gives 84 equals half of a times b, so a times b equals 168. The perimeter gives a plus b plus c equals 56. Using the algebraic identity (a plus b) squared equals a squared plus b squared plus 2ab, we substitute to get (56 minus c) squared equals c squared plus 336. Expanding and solving yields 3136 minus 112c plus c squared equals c squared plus 336, which simplifies to 112c equals 2800. Therefore, the length of the hypotenuse is 25 cm.

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