Math Help - Area of Isosceles Trapezoid

1. Area of Isosceles Trapezoid

Hi there,

How do you find the area of an Isosceles trapezoid if all you have is that the altitude is 15 inches and the diagonal is 25 inches?

2. Aaaaay, The Fonz!

Find the area of an Isosceles trapezoid with altitude 15 and diagonal 25.
Code:
              A       x       B
*---------------*
/:  *            :\
/ :     *         : \
/  :        *      :  \
/   :        25 *   :   \
/  15:              *:    \
/     :               : *   \
/      :               :    * \
D *-------*---------------*-------* C
: 20-x  E       x       F 20-x  :
: - - - -  20 - - - - - :

We have isosceles trapezoid $ABCD$ with diagonal $AC = 25$
. . and altitudes $AE = BF = 15.$

In right triangle $AEC\!:\;\;AE^2 + EC^2 \:=\:AC^2 \quad\Rightarrow\quad 15^2 + EC^2 \:=\:25^2$

. . Hence: . $EC^2 \:=\:400 \quad\Rightarrow\quad EC \:=\:20$

Let $AB \,=\,EF \,=\,x$
. . Then: . $FC\,=\,DE \,=\,20-x$

Hence: . $DC \:=\:(20-x) + x + (20-x) \:=\:40-x$

The area of a trapezoid is: . $A \;=\;\frac{h}{2}(b_1 + b_2)$

We have: . $A \;=\;\frac{AE}{2}(AB + DC) \;=\;\frac{15}{2}\bigg[x + (40-x)\bigg] \;=\;\frac{15}{2}\cdot40 \;=\;\boxed{300}$