Answer
$x = 6 \sqrt {2}$
$y = 6 \sqrt {6}$
Work Step by Step
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 $x$ is the length of the altitude.
Let's plug in our numbers:
$\frac{6}{x} = \frac{x}{12}$
Use the cross products property to get rid of the fractions:
$x^2 = 72$
Rewrite $72$ as the product of a perfect square and another factor:
$x^2 = 36 • 2$
Take the positive square root of each factor to solve for $x$:
$x = 6 \sqrt {2}$
To find $y$, 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.
$\frac{12 + 6}{y} = \frac{y}{12}$
Use the cross products property to get rid of the fractions:
$y^2 = 12(12 + 6)$
Evaluate what is in parentheses first:
$y^2 = 12(18)$
Multiply to simplify:
$y^2 = 216$
Rewrite $216$ as the product of a perfect square and another factor:
$y^2 = 36 • 6$
Take the positive square root of each factor to solve for $y$:
$y = 6 \sqrt {6}$