10.185. In an isosceles trapezoid, the length of the midline is 5, and the diagonals are perpendicular to each other. Find the area of the trapezoid.
Solution.

Let the bases of the trapezoid be $a$ and $b$, and the height be $h$. Then the area 

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-642.jpg?height=352&width=356&top_left_y=160&top_left_x=186)

Fig. 10.153

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-642.jpg?height=498&width=466&top_left_y=95&top_left_x=735)

Fig. 10.154

is $S=\frac{a+b}{2} h$. On the other hand, $S=h^{2}$. Therefore, $h=\frac{a+b}{2}=5$, from which $S=25$.

Answer: 25.