Question

the triangle below is isosceles. find the length of side $x$ in simplest radical form with a rational denominator.

Explanation:

Step1: Apply Pythagorean theorem

In an isosceles right - triangle, if the legs have length $a$ and the hypotenuse has length $c$, by the Pythagorean theorem $c^{2}=a^{2}+a^{2}$. Here $a = 4$, so $x^{2}=4^{2}+4^{2}$.
$x^{2}=16 + 16=32$.

Step2: Solve for $x$

Take the square - root of both sides: $x=\sqrt{32}$.
Simplify $\sqrt{32}=\sqrt{16\times2}=4\sqrt{2}$. Since the denominator is already rational (in this case, there is no denominator in the non - fraction form of the radical expression), the length of side $x$ in simplest radical form with a rational denominator is $4\sqrt{2}$.

Answer:

$4\sqrt{2}$