1979 AMC 12 Problem 21

Question

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

\frac{\pi r}{h+2r}

\frac{\pi r}{h+r}

\frac{\pi r}{2h+r}

\frac{\pi r^2}{h^2+r^2}

none of these

Use the formula

K=rs

for a triangle’s area

For a right triangle, relate the sum of the legs to

h

and

r

If the legs are

x

and

y,

then the right-triangle inradius identity is

x+y=h+2r.

Hence the semiperimeter is

s=\frac{x+y+h}{2}=h+r.

The triangle’s area is

rs=r(h+r),

while the circle’s area is

\pi r^2.

Their ratio is

\frac{\pi r}{h+r}.

Therefore, the correct answer is B.

Problem 21 in other years