8,9

One of the angles of the trapezoid is $30^{\circ}$, and the lines containing the lateral sides of the trapezoid intersect at a right angle. Find the length of the shorter lateral side of the trapezoid if its midline is equal to 10, and one of the bases is equal to 8.

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Draw a line through the vertex of the smaller base of the trapezoid, parallel to the lateral side.

## Solution

The second (larger) base $AD$ of trapezoid $ABCD$, according to the midline theorem, is 12. Through the vertex $C$ of the smaller base $BC$, draw a line parallel to the lateral side $AB$, until it intersects the base $AD$ at point $K$. Triangle $KCD$ is a right triangle ($\angle KCD=90^{\circ}$). Therefore,

$$
AB=KC=\frac{1}{2}KD=\frac{AD-AK}{2}=\frac{AD-BC}{2}=2
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_1b17058527caf5b5d058g-11.jpg?height=341&width=784&top_left_y=2059&top_left_x=637)

## Answer

2.