Calculate the diagonal of a rectangle if the base is 14 less

Calculate the diagonal of a rectangle if the base is 14 less than the height and the diagonal is 22 less than twice the height.

Solution

For a rectangle, the base and height are at right angles, so the diagonal , base and height form a Pythagorean Triplet.

Lets take the height as H. We are given that base the 14 less than the height or H-14.

The diagonal is 22 less than twice the height or 2H-22

Therefore (2H-22)^2 = H^2 + (H-14)^2

=> 4H^2 + 484 - 88H = H^2 + H^2 + 196 - 28H

=> 2H^2 -88H + 28H + 484 - 196 =0

=> 2H^2 - 60H + 288 =0

=> H^2 - 30H + 144 =0

=> H^2 - 24H- 6H + 144= 0

=> H(H - 24) - 6 ( H-24) =0

=> (H - 24)(H - 6) =0

So H can be 24 or 6.

The base of the rectangle is H-14 and it cannot be negative so H= 6 can be eliminated.

Therefore the height is 24, base is 10 and the diagonal is 26