1999 AMC 12 Problem 16

Question

What is the radius of a circle inscribed in a rhombus with diagonals of length

10

and

24?

4

\dfrac{58}{13}

\dfrac{60}{13}

5

6

The diagonals cut the rhombus into four

5

-

12

-

13

right triangles

The inradius equals the altitude from the center to a side of the rhombus

The half-diagonals are

5

and

12,

so each side of the rhombus is

\sqrt{5^2 + 12^2} = 13.

One of the four right triangles formed by the diagonals has legs

5

and

12

and area

30.

The altitude from the center to the side of length

13

is

\dfrac{2 \cdot 30}{13} = \dfrac{60}{13},

which is the inscribed circle’s radius.

Thus, the correct answer is C.

Problem 16 in other years