Task B-3.1. (20 points) The length of the hypotenuse of a right triangle is $c$, and the length of the angle bisector of one of the acute angles is $\frac{c \sqrt{3}}{3}$. What are the lengths of the legs?
First solution. From the image, we have $a=c \cos 2 x=\frac{c \sqrt{3}}{3} \cos x$.

![](https://cdn.mathpix.com/cropped/2024_05_30_9c56249806d755c5106ag-09.jpg?height=411&width=673&top_left_y=962&top_left_x=700)

From the obtained equality, we get the equation $\sqrt{3} \cos 2 x=\cos x$ or after rearranging $2 \sqrt{3} \cos ^{2} x-\cos x-\sqrt{3}=0$.

(6 points)

Its solutions are $\cos x=\frac{\sqrt{3}}{2}$ and $\cos x=-\frac{\sqrt{3}}{3}$ (the second solution does not apply because $x<90^{\circ}$).

From $\cos x=\frac{\sqrt{3}}{2}$, we get the lengths of the legs $\frac{c}{2}$ and $\frac{c \sqrt{3}}{2}$.

(5 points)