# 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 14 , 4 , and 3 , respectively. What is the rectangle's area?

Dec 24, 2017

See a solution process below:

#### Explanation:

The image above is the shapes we are dealing with from the question.

We can use the Pythagorean Theorem to find the length of $B$.

The Pythagorean Theorem states:

${B}^{2} + {C}^{2} = {A}^{2}$ where $A$ is the hypotenuse of a right triangle.

Substituting the values from the problem and solving $B$ gives:

${B}^{2} + {4}^{2} = {14}^{2}$

${B}^{2} + 16 = 196$

${B}^{2} + 16 - \textcolor{red}{16} = 196 - \textcolor{red}{16}$

${B}^{2} + 0 = 180$

${B}^{2} = 180$

$\sqrt{{B}^{2}} = \sqrt{180}$

$B = \sqrt{36 \cdot 5}$

$B = \sqrt{36} \sqrt{5}$

$B = 6 \sqrt{5}$

The formula for the area of a rectangle is:

$A = l \times w$

Substituting the value from the problem and the value we calculated and calculating $A$ gives:

$A = 6 \sqrt{5} \times 3$

$A = 18 \sqrt{5}$

Or

$A = 40.25$ rounded to the nearest hundredth