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

1 Answer
Nov 28, 2017

Condition is xy+8=4x+4y

Explanation:

Area of right angled triangle whose legsarex and y is1/2xy

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