## [ Isosceles, Inscribed, and Circumscribed Trapezoids ]

$\underline{\text { The Midline of a Trapezoid }}$

The area of an isosceles trapezoid circumscribed around a circle is $S$. Find the midline of the trapezoid if the acute angle at its base is $\alpha$.

#
The midline of the given trapezoid is equal to the lateral side

## Solution

Let $x$ be the midline of the given trapezoid. Then the lateral side is also equal to $x$, and the height of the trapezoid is $x$ sin $\alpha$. We have the equation

$$
S=x^{2} \sin \alpha
$$

From this, we find that

$$
x=\sqrt{\frac{S}{\sin \alpha}}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_1b17058527caf5b5d058g-06.jpg?height=421&width=644&top_left_y=1133&top_left_x=707)

## Answer

$\sqrt{\frac{S}{\sin \alpha}}$