1964 AMC 12 Problem 13

Attempt Problem 13 of the 1964 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1964 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

13.

A circle is inscribed in a triangle with side lengths

8,

13,

and

17.

Let the segments of the side of length

8,

made by a point of tangency, be

r

and

s,

with

r\lt s.

What is the ratio

r:s?

1:3

2:5

1:2

2:3

3:4

Compute the semiperimeter of the triangle

A tangency segment adjacent to a vertex equals the semiperimeter minus the opposite side

The semiperimeter is

\frac{8+13+17}{2}=19.

At the endpoints of the side of length

8,

the tangent lengths are

19-13=6

and

19-17=2.

Thus

r:s=2:6=1:3.

Therefore, the correct answer is A.

Problem 13 in other years