[ Area of a trapezoid ]

The lines containing the lateral sides of an isosceles trapezoid intersect at a right angle. Find the sides of the trapezoid if its area is 12, and the height is 2.

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Find the difference and the sum of the bases of the trapezoid.

## Solution

Let $P$ be the point of intersection of the lateral sides $A B$ and $C D$ of trapezoid $A B C D$. Then $A P D$ is an isosceles right triangle, $\angle A=\angle D=45^{\circ}$.

From the vertices of the smaller base $B C$, drop perpendiculars $B M$ and $C K$ to the larger base $A D$. Then $A M=$ $K D=C K=2, A B=C D=2 \sqrt{2}, A D-B C=4$, and since $S_{A B C D}=1 / 2(A D+B C) C K$, then $A D+B C=12$. From the resulting system of equations, we find that $A D=8$ and $B C=4$.

![](https://cdn.mathpix.com/cropped/2024_05_06_04ea0aec75eb28164af1g-21.jpg?height=420&width=866&top_left_y=2420&top_left_x=593)

Answer

Problem