find the length of side $x$ in simplest radical form with a rational denominator.\nanswer attempt 1 out of…

Explanation:

Step1: Identify the triangle type

This is a 45 - 45 - 90 right - triangle. In a 45 - 45 - 90 triangle, the ratio of the sides is $1:1:\sqrt{2}$, where the hypotenuse $c$ and the legs $a$ and $b$ are related by $c = a\sqrt{2}$ (or $c = b\sqrt{2}$). Here, the hypotenuse $c = 4$, and the leg we want to find is $x$.

Step2: Set up the equation

We know that $c=x\sqrt{2}$, and $c = 4$. So, $4=x\sqrt{2}$.

Step3: Solve for $x$

To solve for $x$, we divide both sides of the equation $4=x\sqrt{2}$ by $\sqrt{2}$: $x=\frac{4}{\sqrt{2}}$.

Step4: Rationalize the denominator

Multiply the numerator and denominator by $\sqrt{2}$: $x=\frac{4\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{4\sqrt{2}}{2}=2\sqrt{2}$.

Answer:

$2\sqrt{2}$