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? 

Given:

The perimeter and area of a right-angled triangle are 36 cm and 54 cm2

Formula used:

The perimeter of a right-angled triangle = Base + Perpendicular + Hypotenuse

Area of a right-angled triangle = \(\frac{1}{2} \) × Base × Perpendicular 

Hypotenuse2 = Base2 + Perpendicular2

Calculation:

Let B , P, and H be the base, Perpendicular, and Hypotenuse of the right-angled triangle

B + P + H = 36

B + P = 36 - H      -----(1)

\(\frac{1}{2} × B× P\) = 54

B × P = 108         -----(2)

H2 = B2 + P2

We know that, (a² + b²) = (a + b)² - 2ab

⇒ H² = (B + P)2 - 2BP

Substituting (1) and (2)

⇒ H² = (36 - H)2 - 216

⇒ H2 = 1296 + H2 - 2(36)(H) - 216

⇒ 72 H = 1296 - 216

⇒ 72 H = 1080

⇒ H = 1080/72 = 15 cm

∴ The correct answer is 15 cm.