3. In a right triangle, the height dropped to the hypotenuse is four times shorter than the hypotenuse. Find the acute angles of the triangle.
Answer: 15 and 75 degrees. Solution. Let $A B C$ be the given right-angled triangle with the right angle at vertex $C$. As is known, the median $C M$ is half the hypotenuse $A B$. The height $C H$ is given to be $A B / 4$. Therefore, in the right-angled triangle $C H M$, the hypotenuse $C M$ is twice the length of the leg $C H$. Consequently, the angle $C M H$ is 30 degrees, and the adjacent angle (let's call it $\angle B M C$) is 150 degrees. Since this is the angle at the vertex of the isosceles triangle $B M C$, $\angle A B C = \angle M B C = (180^{\circ} - \angle B M C) / 2 = 15^{\circ}$, from which we obtain the answer.

Grading Guidelines. Answer without justification - 0 points. Referring to the converse of the statement that if a leg is half the hypotenuse, it lies opposite a 30-degree angle - -3 points. Trigonometric solution without justification of the values of the necessary functions for angles of 15 and 71 degrees - no more than 3 points.