#### Answer

$x = 5\sqrt 2$

#### Work Step by Step

In a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle, the hypotenuse is $\sqrt 2$ times each leg. Let's write an equation to solve for $x$, one of the legs:
$10 = \sqrt 2(x)$
Divide each side of the equation by $\sqrt 2$ to solve for $x$:
$x = \frac{10}{\sqrt 2}$
To simplify this fraction, we multiply both the numerator and denominator by the denominator:
$x = \frac{10}{\sqrt 2} • \frac{\sqrt 2}{\sqrt 2}$
Multiply to simplify:
$x = \frac{10\sqrt 2}{2}$
Divide the numerator and denominator by their greatest common factor, $2$:
$x = 5\sqrt 2$