Problem: The length of the hypotenuse of a right triangle is h, and the radius of the inscribed circle is r. The ratio of the area of the circle to the area of the triangle is
Answer Choices:
A. h+2rΟrβ
B. h+rΟrβ
C. 2h+rΟrβ
D. h2+r2Οr2β
E. none of these
Solution:
In the adjoining figure, x and y are the lengths of the legs of the triangle, so that h=(yβr)+(xβr)
=x+yβ2r,
x+y=h+2r;
x2+y2=h2.
The area of the triangle ABC is
21βxy=21β[2(x+y)2β(x2+y2)β]=41β[(h+2r)2βh2]=hr+r2
Thus the desired ratio is hr+r2Οr2β=h+rΟrβ.
