# The width of a rectangle is 5 cm and the length of its diagonal is 13 cm. How long is the other side of the rectangle and what is the area?

Oct 28, 2016

The length of the rectangle is $12 c m$ and the area of the rectangle is $60 c {m}^{2}$.

#### Explanation:

By definition, the angles of a rectangle are right. Therefore, drawing a diagonal creates two congruent right triangles. The diagonal of the rectangle is the hypotenuse of the right triangle. The sides of the rectangle are the legs of the right triangle. We can use the Pythagorean Theorem to find the unknown side of the right triangle, which is also the unknown length of the rectangle.

Recall that the Pythagorean Theorem states that the sun of the squares of the legs of a right triangle is equal to the square of the hypotenuse. ${a}^{2} + {b}^{2} = {c}^{2}$

${5}^{2} + {b}^{2} = {13}^{2}$
$25 + {b}^{2} = 169$
$25 - 25 + {b}^{2} = 169 - 25$
${b}^{2} = 144$
$\sqrt{{b}^{2}} = \sqrt{144}$
$b = \pm 12$

Since the length of the side is a measured distance, the negative root is not a reasonable result. So the length of the rectangle is $12$ cm.

The area of a rectangle is given by multiplying the width by the length.

$A = \left(5 c m\right) \left(12 c m\right)$
$A = 60 c {m}^{2}$