question find the length of side x in simplest radical form with a rational denominator.

Explanation:

Step1: Identify the trigonometric ratio

Since it's a right - triangle and we know an angle of 45° and the adjacent side to the 45° angle is 4, and we want to find the hypotenuse $x$. We use the cosine function. $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 45^{\circ}$, and $\cos45^{\circ}=\frac{\sqrt{2}}{2}$, adjacent side $a = 4$, and hypotenuse $x$. So, $\cos45^{\circ}=\frac{4}{x}$.

Step2: Solve for $x$

We have $\frac{\sqrt{2}}{2}=\frac{4}{x}$. Cross - multiply to get $x\sqrt{2}=8$. Then $x=\frac{8}{\sqrt{2}}$.

Step3: Rationalize the denominator

Multiply the numerator and denominator by $\sqrt{2}$: $x=\frac{8\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{8\sqrt{2}}{2}=4\sqrt{2}$.

Answer:

$4\sqrt{2}$