Answer
$y = 3 \sqrt 3$
$x = 6 \sqrt 3$
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:
The geometric mean of two numbers can be found using the following 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:
$\frac{3}{y} = \frac{y}{9}$
Use the cross products property to get rid of the fractions:
$y^2 = 27$
Rewrite $48$ as the product of a perfect square and another factor:
$y^2 = 9 • 3$
Take the positive square root of each factor to solve for $y$:
$y = 3 \sqrt 3$
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.
$\frac{9 + 3}{x} = \frac{x}{9}$
Use the cross products property to get rid of the fractions:
$x^2 = 12 • 9$
Multiply to simplify:
$x^2 = 108$
Rewrite $108$ as the product of a perfect square and another factor:
$x^2 = 36 • 3$
Take the positive square root of each factor to solve for $x$:
$x = 6 \sqrt 3$
