10.361 In triangle $A B C$, the measure of angle $A$ is twice the measure of angle $B$, and the lengths of the sides opposite these angles are 12 cm and 8 cm, respectively. Find the length of the third side of the triangle.
Solution.
A

![](https://cdn.mathpix.com/cropped/2024_05_21_f024bf2ff7725246f3bfg-225.jpg?height=286&width=213&top_left_y=611&top_left_x=236)
Let $A K$ be the bisector of $\angle A$. Then
$\angle K A C = \angle A B C \Rightarrow \triangle A K C \sim \triangle A B C \Rightarrow$
$\Rightarrow \frac{A C}{B C} = \frac{K C}{A C} = \frac{A K}{A B} \Rightarrow K C = \frac{A C^2}{B C} = \frac{16}{3}$,
$A K = B K = B C - K C = 12 - \frac{16}{3}$, $A B = \frac{A K \cdot A C}{K C} = 10$.

Answer: 10 cm