Question

A trapezoid has a top base of 32 and a bottom base of 22. Its legs are each 13. What is its area?

Answer 1

`324`

Explanation:

The area of a trapezoid is the average of its bases times its height. This can be expressed as `A_("trapezoid")=frac(h(b_1+b_2))(2)`.

We have the two bases, but we don't have the height. Draw the trapezoid on a piece of paper.

Now, write in the measurements. If both legs are `13`, and the bases are `22` and `32`, how can we find the height?

We can use the Pythagorean theorem. If one base is `22` and the other is `32`, the long base is `10` longer than the short one. If we have an isosceles trapezoid like in the picture, the long base will extend `5` more in either direction. So, if you drew a line cutting straight down from the vertex where the SHORT base and the height met all the way to the LONG base, you will have a right triangle with a hypotenuse of `13` and a leg of `5`.

We can use the Pythagorean theorem to figure out that the other leg of the right triangle, which is also the height of the trapezoid, has length `12`.

Then, we plug our height and base lengths back into the area formula to determine that the area of the trapezoid is `color(red)(324`.