Question

The perimeter of right triangle is 144 cm. Its hypotenuse is 65 cm. Find the length of the legs of this triangle.

Answer 1

Let a, b and c be the sides of the triangle. Then, a+b+c = 144. If c is hypotenuse, 65cm, then

a+b+65 = 144

or a+b = 144 - 65 = 79 -------- (1)

For a right-angled triangle, a^2+b^2 = c^2

or a^2+b^2 = 65^2      --------- (2)

Squaring (1), we get

(a+b)^2 = 79^2

or a^2+b^2 +2ab = 79^2

But a^2+b^2 = 65^2

Thus, we get 65^2+2ab = 79^2 or

2ab = 79^2-65^2 = (79+65)*(79-65) = 144*14 = 2016

or ab = 2016/2 = 1008.

We can rewrite this as b = 1008/a.

Let us substitute the value of b in (1). We get

a+1008/a = 79

or a^2+1008=79a

or a^2-79a+1008 = 0

Thus, a = (79±sqrt(79^2-4*1*1008))/2*1 = (79±sqrt(6241-4032))/2

or a = (79±sqrt(2209))/2 = (79±47)/2

a = 126/2 = 63 or a = 32/2 = 16.

Substituting the value of a = 63 in b = 1008/a = 1008/63 = 16

Substituting the value of a = 16 in b = 1008/a = 1008/16 = 63

Thus, the other two sides are 16 and 63 respectively.