$\frac{[\text { Similar triangles (other) })]}{[\underline{\text { Trapezoids (other) }}]}$

In trapezoid $A B C D$, side $A B$ is perpendicular to the bases $A D$ and $B C$. Point $E$ is the midpoint of side $C D$.

Find the ratio $A D: B C$, if $A E=2 A B$ and $A E \perp C D$.
Let $K$ be the projection of vertex $C$ onto $A D$. Consider similar triangles $C K D$ and $A E D$.

## Solution

Let $K$ be the projection of vertex $C$ onto the base $A D$. Triangles $C K D$ and $A E D$ are similar with a ratio of ${ }^{C K} / A E=$ $A B / A E=1 / 2$, therefore

$K D=1 / 2 E D=1 / 4 C D=1 / 8 A D, B C=A K=7 / 8 A D$.

![](https://cdn.mathpix.com/cropped/2024_05_06_96d74c2697cd905bd948g-17.jpg?height=366&width=671&top_left_y=1941&top_left_x=705)

## Answer

$8: 7$.

Submit a comment