$\left[\begin{array}{l}{[\text { Inscribed quadrilaterals }]} \\ \text { Area of a trapezoid }]\end{array}\right.$

The midline of an isosceles trapezoid is 5. It is known that a circle can be inscribed in the trapezoid.

The midline of the trapezoid divides it into two parts, the ratio of the areas of which is $\frac{7}{13}$. Find the height of the trapezoid.
Solve the system of equations with two unknowns - the lengths of the bases of the trapezoid.

## Solution

Let $x$ and $y$ be the bases of the trapezoid. Then

$$
\left\{\begin{array}{l}
x+y=10 \\
\frac{x+5}{y+5}=\frac{7}{13}
\end{array}\right.
$$

From this, we find that $x=2$ and $y=8$. Therefore, the lateral side of the trapezoid is 10.

Drop a height from the vertex of the upper base to the lower base and find it using the Pythagorean theorem.

![](https://cdn.mathpix.com/cropped/2024_05_06_93426a107eaf3f76289cg-35.jpg?height=389&width=640&top_left_y=503&top_left_x=709)

## Answer

4.