
Problem 6.
A circle is inscribed in a right triangle with dimensions x, y, and r
(x and y are the distances from the corners to where the circle intersects
the legs and r is the radius of the circle) as shown below.
Find the area of the triangle in terms of x and y only (without r).
Solution:
Let A be the area of the triangle.
Express twice the area in two different ways:
(1) Use the formula for twice a triangle: 2A = base * height
(2) Add the components to form the area of the rectangle with
dimensions
(y+r) and (x+r):
Therefore,
Notice that the left side of the last equation is half the right
side of the equation
and is therefore the area of the triangle.


