Question

The perimeter of a right triangle is 144cm and it's hypotenuse measure 65 cm . Find the length of other two sides and calculate it's area verify the results using Herons formula

Answer 1

The perimeter of right angled triangle is 144 cm.

hypotenuous is 65 cm .

let the two sides are x and y .

x 2 + y2 = 652

perimeter of right angled triangle is 144 cm.

And x + y + 65 = 144

x + y = 144 - 65 = 79

squaring both sides, we get

( x + y )2 = 792

=> x2 + y2 + 2 x y = 6241

4225 + 2 x y = 6241

2 x y = 6241 - 4225

= 2016

x y = 2016 / 2

= 1008

y = 1008 / x

sub: y = 1008 / x in x + y = 79 , we get

x + 1008 / x = 79

x2 + 1008 = 79 x

x2 - 79 x +1008 = 0

x2 - 63 x - 16 x + 1008 = 0

x ( x - 63 ) - 16 ( x - 63 ) =0

( x - 63 ) ( x - 16 ) = 0

so, x = 63 cm or x = 16 cm

the length of other sides are 63 cm , 16 cm.

Area of right angled tringle is

1 / 2 x 63 x 16 = 504 cm^2

Verify the result using Heron's Formula

S = ( 16 + 63 + 65 ) / 2 = 72

A = ( s x ( s-a ) x ( s - b ) x ( s - c ) )1/2

= ( 72 x (72 - 16 ) x ( 72 - 63 ) x ( 72 - 65 ) )1/2

= ( 72 x 56 x 9 x 7 ) 1/2

= ( 9 x 8 x 8 x 7 x 9 x 7 )1/2

= 9 x 8 x 7 = 504 cm^2

Result is verified.