5. The height of a right triangle, dropped to its hypotenuse, divides the bisector of an acute angle in the ratio 5 : 2, counting from the vertex. Find the measure of this angle.

Answer: $\arccos \frac{5}{7}$.
Solution. Let $A D=5 x, D L=2 x$. Then $\cos \angle L A H=$ $=\frac{A H}{A D}=\frac{A H}{5 x}=\cos \angle L A C=\frac{A C}{A L}=\frac{A C}{7 x}$. Therefore,

![](https://cdn.mathpix.com/cropped/2024_05_06_2cf4ae7b738ff8febec4g-2.jpg?height=334&width=642&top_left_y=176&top_left_x=1301)
$\cos \angle C A H=\frac{A H}{A C}=\frac{5}{7}$.