Question

The sides of a right triangle are such that the longest side is 4 m more than the shortest side and the third side is 2 m less than the longest side. Find the length of each side of the triangle. Also, find the difference between the numerical values of the area and the perimeter of the given triangle.

Answer 1

Let the length of the shortest side be x m.

Longer side = x + 4 m

Third side = x + 4 − 2

= x + 2m

By applying the Pythagorean theorem,

AC² = AB² + BC²

(x + 4)² = (x + 2)² + x²

x² + 16 + 8x = x² + 4 + 4x + x²

2x² + 4x + 4 − x² − 8x − 16 = 0

x² − 4x − 12 = 0

x² − 6x + 2x − 12 = 0

x(x − 6) + 2(x − 6) = 0

(x − 6) (x + 2) = 0

x − 6 = 0 or x + 2 = 0

x = 6, x = −2     ....(∵ Length can not be negative.)

x = 6, x ≠ −2

∴ Sides of a right-angled triangle are:

BC = 6 cm

AB = 6 + 2 = 8 m

AC = 6 + 4 = 10 m

Area = `1/2 xx "Base" xx "Height"`

= `1/2 xx 6 xx 8`

= 24 m²

Perimeter = 6 + 8 + 10

= 24 m

Difference between Area and Perimeter = 24 − 24

= 0