Answer
$x = 5 \sqrt 2$
Work Step by Step
In a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle, we know that the hypotenuse is $\sqrt 2$ times each leg. Let's write an equation to solve for $x$, the length of a leg:
$10 = \sqrt 2(x)$
Divide each side by $\sqrt 2$ to solve for $x$:
$x = \frac{10}{\sqrt 2}$
To simplify this fraction, we need to get rid of the radical in the denominator by multiplying both the numerator and denominator by the denominator:
$x = \frac{10}{\sqrt 2} • \frac{\sqrt 2}{\sqrt 2}$
Multiply:
$x = \frac{10 \sqrt 2}{\sqrt 4}$
Take the square root of the denominator:
$x = \frac{10 \sqrt 2}{2}$
Divide the numerator and denominator by their greatest common factor:
$x = 5 \sqrt 2$
