# A right triangle has sides A, B, and C. Side A is the hypotenuse and side B is also a side of a rectangle. Sides A, C, and the side of the rectangle adjacent to side B have lengths of 7 , 6 , and 15 , respectively. What is the rectangle's area?

Feb 18, 2016

$15 \sqrt{13}$

#### Explanation:

The above (not to scale) picture contains the information given in the problem. In many geometry problems, drawing a picture is a good way to start.

The area of the rectangle is the product of the lengths of its sides, in this case $15 B$. To solve for $B$, we can use the Pythagorean theorem, which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of its legs. In this case, that translates to ${A}^{2} = {B}^{2} + {C}^{2}$

Substituting in the given values for $A$ and $C$, we obtain

$49 = {B}^{2} + 36$

$\implies {B}^{2} = 13$

$\implies B = \sqrt{13}$

Thus the area of the rectangle is $15 \sqrt{13}$