The perimeter of a right triangle is 40 cm and its hypotenuse measures 17 cm. Find the area of the triangle.

Given: Perimeter = 40 cm

Hypotenuse = 17 cm

The diagram is given as:

Let the sides be a, b and c(hypotenuse).

Therefore, a + b + c = 40 cm

⇒ a + b + 17 = 40 cm

⇒ a + b = 40 - 17 cm

⇒ a + b = 23 cm

⇒ a = (23-b) cm

Now we know that,

Base² + Perpendicular² = Hypotenuse²

⇒ a² + b² = c²

⇒ (23-b)² + b² = 17²

⇒ 23² + b²-46b + b² = 289

⇒ 529 + b²-46b + b² = 289

⇒ 2b²-46b + 240 = 0

⇒ b²-23b + 120 = 0

⇒ b²-8b-15b + 120 = 0

⇒ b(b-8)-15(b-8) = 0

⇒ (b-8)(b-15) = 0

This gives us two equations,

i. b-8 = 0

⇒ b = 8

ii. b-15 = 0

⇒ b = 15

Let b = 8 cm

⇒ a = (23-b) cm

⇒ a = (23-8) cm

⇒ a = 15 cm

Now,

Area of triangle = 1/2 × base × height

= 1/2 × 8 × 15

= 60 cm²