Question

A circle of radius 2 inscribed in a right triangle. Find the simplest expression possible for the triangle's area as a function of its hypotenuse.

Answer 1

Hmmm. If the legs are a and b, and the hypotenuse is c, then we have

r = (a+b-c)/2 = ab/(a+b+c)

Since A = ab/2, that means

r = 2A / (a+b+c)

with r=2, that means

A = (a+b+c)

But, since a+b+c = (a+b-c)+2c, we have

A = 2r+2c
A = 4+2c