Hello, zachcumer!
Code:
D * * * * * * * C
* * *
* * *
12 * * 20 * 15
* * *
* * *
A * * * * * * * B
Area of a trapezoid: .$\displaystyle A \;=\;\frac{h}{2}(b_1 + b_2)$
We know that: $\displaystyle h = AD = 12$
. . But what are $\displaystyle b_1 = CD\text{ and }b_2 = AB$ ?
In right triangle $\displaystyle CDA\!:\;\;CD^2 + AD^2 \:=\:AC^2\quad\Rightarrow\quad CD^2 + 12^2 \:=\:20^2$
. . $\displaystyle CD^2 \:=\:256\quad\Rightarrow\quad \boxed{CD = 16}$
In right triangle $\displaystyle ACB\!:\;\;AB^2 \:=\:AC^2 + BC^2\quad\Rightarrow\quad AB^2 \:=\:20^2 + 15^2$
. . $\displaystyle AB^2 \:=\:625\quad\Rightarrow\quad \boxed{AB \:=\:25}$
Therefore: .$\displaystyle A \;=\;\frac{12}{2}(16 + 25) \;=\;246$