Multiple choice

The perimeter and the area of a right-angled triangle are 36 cm and 54 square cm respectively. What is the length of the hypotenuse?

  1. 12 cm

  2. 14 cm

  3. 15 cm

  4. 16 cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Perimeter = a+b+c = 36. Area = 0.5ab = 54 => ab = 108. In a right triangle, a^2+b^2=c^2. (a+b)^2 = a^2+b^2+2ab = c^2+2(108) = c^2+216. Also a+b = 36-c. (36-c)^2 = c^2+216 => 1296 - 72c + c^2 = c^2 + 216 => 72c = 1080 => c = 15.

AI explanation

Let the legs be a and b, and the hypotenuse be c. The area gives ab/2 = 54 so ab = 108, and the perimeter gives a + b + c = 36. Using the algebraic identity a² + b² = (a + b)² - 2ab, we can substitute the Pythagorean theorem c² = a² + b² to get c² = (36 - c)² - 216. Expanding and solving c² = 1296 - 72c + c² - 216 yields 72c = 1080, so c = 15. The length of the hypotenuse is 15 cm.

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