## Geometry: Common Core (15th Edition)

$x = 9$
We know that each leg of the triangle is the geometric mean of the hypotenuse and the hypotenuse that is adjacent to that leg: $\frac{a}{y} = \frac{y}{b}$, where $a$ and $b$ are the length of the hypotenuse and the length of the segment of the hypotenuse closest to the leg, and $y$ is the length of the leg. $\frac{16 + x}{15} = \frac{15}{x}$ Use the cross products property to get rid of the fractions: $x(16 + x) = 225$ Use the distributive property: $16x + x^2 = 225$ Subtract $225$ from each side of the equation to set up a quadratic equation: $x^2 + 16x - 225 = 0$ Factor the quadratic equation: $(x + 25)(x - 9) = 0$ Set each factor equal to $0$: $x + 25 = 0$ or $x - 9 = 0$ Add or subtract to solve for $x$: $x = -25$ or $x = 9$ We discard the negative result because length cannot be negative: $x = 9$