Question

What conditions must hold for a right triangle with legs x and y to have its perimeter numerically equal to its area?

Answer 1

Condition is `xy+8=4x+4y`

Explanation:

Area of right angled triangle whose legsare`x` and `y` is`1/2xy`

and its perimeter is `x+y+sqrt(x^2+y^2)`

if `x+y+sqrt(x^2+y^2)=1/2xy`

or `sqrt(x^2+y^2)=(xy)/2-(x+y)=(xy-2x-2y)/2`

or `x^2+y^2=1/4(4x^2+4y^2+x^2y^2+8xy-4x^2y-4xy^2)`

or `4x^2+4y^2=4x^2+4y^2+x^2y^2+8xy-4x^2y-4xy^2`

or `x^2y^2+8xy=4x^2y+4xy^2`

and dividing by `xy` as it is non-zero, we get

`xy+8=4x+4y`