Question

What is the radius of a circle inscribed in a rhombus with diagonals of length $10$ and $24?$

$4$

$\dfrac{58}{13}$

$\dfrac{60}{13}$

$5$

$6$

Answer: C

Concepts:rhombus incircle, incenter, and inradius Pythagorean Theorem triangle area

Difficulty rating: 1610

Solution:

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.

1999 AMC 12 Problem 16

Answer: C

Difficulty rating: 1610

Solution:

Problem 16 in Other Years