Answer
$x = \frac{50}{3}$
Work Step by Step
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. We can use this information to set up a proportion to find $x$:
$\frac{a}{x} = \frac{x}{b}$, 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.
Let's plug in what we know:
$\frac{(x - 6) + 6}{10} = \frac{10}{6}$
Evaluate what is inside the brackets first:
$\frac{x}{10} = \frac{10}{6}$
Use the cross products property to get rid of the fractions:
$6x = 100$
Divide each side by $6$ to solve for $x$:
$x = \frac{100}{6}$
Simplify the fraction by dividing both the numerator and denominator by their greatest common factor, $2$:
$x = \frac{50}{3}$
