[ $\underline{\text { Projections of the bases, sides, or vertices of a trapezoid ] }}$ [ Pythagorean Theorem (direct and inverse) ]

The lateral side of an isosceles trapezoid is 41, the height is 40, and the midline is 45. Find the bases.
Denote the bases by $a$ and $b$ and set up a system of equations.

## Solution

Drop perpendiculars $B M$ and $C N$ from vertices $B$ and $C$ of the smaller base $B C$ of trapezoid $A B C D$ to the larger base. Let $B C=a, A D=b$. By the Pythagorean theorem, $A M^{2}=41^{2}-40^{2}=81$. From the system $a+b=90, b-a=18$, we find that $a=36, b=54$.

![](https://cdn.mathpix.com/cropped/2024_05_06_4fce0705b02fa9e106d4g-23.jpg?height=306&width=500&top_left_y=1&top_left_x=779)

## Answer

36 and 54.