The area of an isosceles trapezoid is 32. The cotangent of the angle between the diagonal and the base is 2. Find the height of the trapezoid.

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The projection of the diagonal of an isosceles trapezoid onto its base is equal to the midline of the trapezoid.

## Solution

Let $h$ be the height we are looking for. The projection of the diagonal of the trapezoid onto its larger base is $2h$. Therefore, the midline of the trapezoid is also $2h$. According to the problem, $2h^2 = 32$, from which we get $h = 4$.

![](https://cdn.mathpix.com/cropped/2024_05_06_837b6360ec703a44a031g-25.jpg?height=229&width=644&top_left_y=729&top_left_x=707)

## Answer

4.