# 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 2 , 2/3 , and 3 , respectively. What is the rectangle's area?

Mar 1, 2016

$4 \sqrt{2}$

#### Explanation:

The area of a rectangle is $A = B a s e \cdot H e i g h t$ The base is given as $3$ and the height is $B$. We can use Pythagoras' Theorem to calculate $B$ because it is a right triangle and we have been given the lengths of the other two sides.

${A}^{2} = {B}^{2} + {C}^{2}$
${B}^{2} = {A}^{2} - {C}^{2} = {2}^{2} - {\left(\frac{2}{3}\right)}^{2}$
${B}^{2} = 4 - \frac{4}{9} = \frac{32}{9}$

$B = \sqrt{\frac{32}{9}} = \frac{4 \sqrt{2}}{3}$

Area $A = 3 \cdot \frac{4 \sqrt{2}}{3} = 4 \sqrt{2}$