Question

The perimeter of a right-angled triangle is five times the length of the shortest side. The numerical value of the area of the triangle is 15 times the numerical value of the length of the shortest side. Find the lengths of the three sides of the triangle.

Answer 1

Let the shortest side of the right-angled triangle be x.

Let the other side be y and the hypotenuse be z.

Given:

Perimeter = 5 × shortest side

x + y + z = 5x

⇒ y + z = 4x     ...(1)

Area = 15 × shortest side

Area of a right-angled triangle:

`1/2 xx x xx y = 15x`

Cancel x (x ≠ 0):

`1/2 y = 15`

⇒ y = 30

z² = x² + y²

= x² + 30²

= x² + 900

From (1):

z = 4x − y = 4x − 30

(4x − 30)² = x² + 900

16x² − 240x + 900 = x² + 900

15x² − 240x = 0

15x(x − 16) = 0

x = 16

Find the remaining sides:

y = 30

`z = sqrt(16^2 + 30^2)`

`= sqrt(256 + 900)`

`= sqrt1156`

= 34

Lengths of the sides are:

Shortest side = 16

Other side = 30

Hypotenuse = 34