8,9

In triangle $ABC$, a line $BD$ is drawn such that $\angle ABD = \angle C$. Find the segments $AD$ and $DC$, if $AB = 2$ and $AC = 4$.
Triangles $A B D$ and $A C B$ are similar.

## Solution

Triangles $A B D$ and $A C B$ are similar by two angles. Therefore, ${ }^{A D} / A B={ }^{A B} / A C$. Consequently, $A D=\left({ }^{A B} / A C\right)^{2}=1$, $C D=A C-A D=4-1=3$.

![](https://cdn.mathpix.com/cropped/2024_05_06_0ef0b80dacb6d98eff5ag-31.jpg?height=274&width=666&top_left_y=1232&top_left_x=702)

## Answer

## 1 and 3.