# The diagonal of a rectangle is 13 inches. The length of the rectangle is 7 inches longer than its width. How do you find the length and width of the rectangle?

Oct 15, 2015

Let's call the width $x$. Then the length is $x + 7$

#### Explanation:

The diagonal is the hypotenuse of a rectangular triangle.
So:
${d}^{2} = {l}^{2} + {w}^{2}$ or (filling in what we know)

${13}^{2} = 169 = {\left(x + 7\right)}^{2} + {x}^{2} = {x}^{2} + 14 x + 49 + {x}^{2} \to$

$2 {x}^{2} + 14 x - 120 = 0 \to {x}^{2} + 7 x - 60 = 0$

A simple quadratic equation resolving into:

$\left(x + 12\right) \left(x - 5\right) = 0 \to x = - 12 \mathmr{and} x = 5$

Only the positive solution is useable so:

$w = 5 \mathmr{and} l = 12$

Extra :
The (5,12,13) triangle is the second-simplest Pythagorean triangle (where all sides are whole numbers). The simplest is (3,4,5). Multiples likes (6,8,10) don't count.