## Geometry: Common Core (15th Edition)

$y = 4 \sqrt {5}$ $x = 12$
The altitude of a right triangle is the geometric mean of the lengths of the two hypotenuse segments. Let's set up that proportion: $\frac{a}{y} = \frac{y}{b}$, where $a$ and $b$ are the lengths of the two hypotenuse segments and $y$ is the length of the altitude. Let's plug in our numbers: $\frac{8}{y} = \frac{y}{10}$ Use the cross products property to get rid of the fractions: $y^2 = 80$ Rewrite $80$ as the product of a perfect square and another factor: $y^2 = 16 • 5$ Take the positive square root of each factor to solve for $y$: $y = 4 \sqrt {5}$ To find $x$, we know that each leg of the triangle is the geometric mean of the hypotenuse and the hypotenuse segment that is adjacent to that leg: $\frac{a}{x} = \frac{x}{b}$, where $a$ and $b$ are the length of the hypotenuse and the length of the segment of the hypotenuse closest to the leg, and $x$ is the length of the leg. $\frac{18}{x} = \frac{x}{8}$ Use the cross products property to get rid of the fractions: $x^2 = 144$ Take the positive square root to solve for $x$: $x = 12$