Question

find the side labeled x. x =

Explanation:

Step1: Identify trigonometric relation

In a right - triangle, we use the cosine function. $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 45^{\circ}$, the adjacent side to the $45^{\circ}$ angle is the side of length 40, and the hypotenuse is $x$.
So, $\cos45^{\circ}=\frac{40}{x}$.

Step2: Recall cosine value

We know that $\cos45^{\circ}=\frac{\sqrt{2}}{2}$. Then $\frac{\sqrt{2}}{2}=\frac{40}{x}$.

Step3: Solve for $x$

Cross - multiply to get $x\times\sqrt{2}=2\times40 = 80$. Then $x=\frac{80}{\sqrt{2}}$. Rationalize the denominator: $x=\frac{80\sqrt{2}}{2}=40\sqrt{2}$.

Answer:

$40\sqrt{2}$