# Problem 5. (3 points)

The height of a rhombus, drawn from the vertex of its obtuse angle, divides the side of the rhombus in the ratio $1: 3$, counting from the vertex of its acute angle. What part of the area of the rhombus is the area of the circle inscribed in it?

Answer: $\frac{\pi \sqrt{15}}{16}$
Solution:

The problem is solved up to similarity, so we can assume that the side of the rhombus is 4. Then the height divides it into segments of 1 and 3.

The height of the rhombus can be found using the Pythagorean theorem: $\sqrt{4^{2}-1^{2}}=\sqrt{15}$. This height is also the diameter of the circle.

From this, we get the ratio of the areas: $\frac{\left(\frac{\sqrt{15}}{2}\right)^{2} \pi}{4 \sqrt{15}}=\frac{\pi \sqrt{15}}{16}$