$\left[\begin{array}{l}\text { Transfer of side, diagonal, etc. } \\ \text { Area of a trapezoid }\end{array}\right]$

In an isosceles trapezoid, the midline is equal to $a$, and the diagonals are perpendicular to each other. Find the area of the trapezoid.
Through one of the vertices of the smaller base of the trapezoid, draw a line parallel to the diagonal.

## Solution

Through vertex $C$ of the smaller base $B C$ of the given trapezoid $A B C D$, draw a line parallel to diagonal $B D$, until it intersects line $A D$ at point $K$.

Triangle $A C K$ is a right isosceles triangle. Its hypotenuse

$$
A K=A D+D K=A D+B C=2 a,
$$

and the height drawn to the hypotenuse is $a$. The area of the given trapezoid is equal to the area of triangle $A C K$, i.e.,

$$
S_{\mathrm{ABCD}}=S_{\triangle \mathrm{ACK}}=\frac{1}{2} \cdot 2 a \cdot a=a^{2}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_b9b3326e3456e69a696eg-19.jpg?height=351&width=734&top_left_y=1711&top_left_x=662)

## Answer

$a^{2}$