#### Answer

$x = 9$

#### Work Step by Step

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.
$\frac{16 + x}{15} = \frac{15}{x}$
Use the cross products property to get rid of the fractions:
$x(16 + x) = 225$
Use the distributive property:
$16x + x^2 = 225$
Subtract $225$ from each side of the equation to set up a quadratic equation:
$x^2 + 16x - 225 = 0$
Factor the quadratic equation:
$(x + 25)(x - 9) = 0$
Set each factor equal to $0$:
$x + 25 = 0$ or $x - 9 = 0$
Add or subtract to solve for $x$:
$x = -25$ or $x = 9$
We discard the negative result because length cannot be negative:
$x = 9$