## Geometry: Common Core (15th Edition)

$y = 2 \sqrt {21}$ $x = 10$
The altitude of a right triangle is the geometric mean of the lengths of the two hypotenuse segments. Let's set up that proportion: $\frac{a}{x} = \frac{x}{b}$, 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: $\frac{21}{y} = \frac{y}{4}$ Use the cross products property to get rid of the fractions: $y^2 = 84$ Rewrite $84$ as the product of a perfect square and another factor: $y^2 = 4 • 21$ Take the positive square root of each factor to solve for $y$: $y = 2 \sqrt {21}$ To find $x$, we know that each leg of the triangle is the geometric mean of the hypotenuse and the hypotenuse that is adjacent to that leg: $\frac{a}{x} = \frac{x}{b}$, 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. $\frac{21 + 4}{x} = \frac{x}{4}$ Use the cross products property to get rid of the fractions: $x^2 = 25 • 4$ Multiply to simplify: $x^2 = 100$ Take the positive square root to solve for $x$: $x = 10$