## Task A-3.7. (10 points)

In triangle $A B C$, the length of side $\overline{B C}$ is 6, the cosine of the angle at vertex $B$ is $\frac{4}{5}$, and the length of the radius of the inscribed circle is 1. Determine the lengths of sides $\overline{A B}$ and $\overline{A C}$ of this triangle.
## First Solution.

Let $|BC|=a, |AC|=b, |AB|=c, \angle ABC=\beta$ and let $r$ be the radius of the inscribed circle of triangle $ABC$. We have

$$
P(ABC)=\frac{1}{2} r(a+b+c)=\frac{1}{2} a c \sin \beta
$$

(*) 2 points

Since $\cos \beta=\frac{4}{5}$, we have $\sin \beta=\frac{3}{5}$, and given $a=6$, from the equality $(*)$ we get

$1 \cdot(6+b+c)=6 \cdot c \cdot \frac{3}{5}$, i.e., $b+6=\frac{13}{5} c$.

2 points

According to the cosine rule, we have

$$
b^{2}=a^{2}+c^{2}-2 a c \cos \beta=36+c^{2}-\frac{48}{5} c
$$

From the last two equalities, we get $c=b=\frac{15}{4}$.

4 points