Question

The sides ( other than hypotenuse) of a right triangle are in the ratio is 6 : 8 . A rectangle is inscribed on its hypotenuse , the hypotenuse being the longer side of the rectangle. The breadth of the rectangle is 4/5 th of its length. Find the shortest side of the right triangle , if the perimeter of the rectangle is 180 cm.

Answer 1

let length of rectangle be x

breadth is 4/5 x

perimeter of rectangle = 180 = 2(l + b)

$2(x + \frac{4}{5} x) = 180 \\ x + \frac{4}{5} x = 90 \\ 5x + 4x = 90 \times 5 \\ 9x = 450 \\ x = \frac{450}{9} = 50 \\ lenght \: of \: rectangle \: = 50cm \: which \: is \: equal \: to \: hypotenuse \: of \: triangle$sides of triangle are in ratio of 6:8

6x and 8x respectively

by pythagoras theorem

${50}^{2} = {(6x)}^{2} + {(8x)}^{2} \\ 2500 = 36 {x}^{2} + 64 {x}^{2} \\ 2500 = 100 {x}^{2} \\ {x}^{2} = \frac{2500}{100} \\ {x}^{2} = 25 \\ x = \sqrt{25} = \\ x = 5 \\ sides \: of \: triangle \: 6x = 6 \times 5 = 30 \\ 8x = 8 \times 5 = 40$

therefore the shortest side of triangle is 30cm