Question

4.
there is a right triangle in the picture. the right angle is at the bottom - left vertex. the hypotenuse is labeled 4, and one of the acute angles is 45 degrees. the side opposite the 45 - degree angle (the vertical leg) is labeled x.

Explanation:

Step1: Identify triangle type

This is a right - isosceles triangle (one angle \(45^{\circ}\), right angle \(90^{\circ}\), so the third angle is also \(45^{\circ}\)), so the two legs are equal, and we can use trigonometric ratios. Let's use the sine function: \(\sin(45^{\circ})=\frac{\text{opposite}}{\text{hypotenuse}}\). The opposite side to the \(45^{\circ}\) angle is \(x\), and the hypotenuse is \(4\).

Step2: Recall \(\sin(45^{\circ})\) value

We know that \(\sin(45^{\circ})=\frac{\sqrt{2}}{2}\).

Step3: Solve for \(x\)

From \(\sin(45^{\circ})=\frac{x}{4}\), substitute \(\sin(45^{\circ})=\frac{\sqrt{2}}{2}\) into the equation: \(\frac{\sqrt{2}}{2}=\frac{x}{4}\). Cross - multiply to get \(x = 4\times\frac{\sqrt{2}}{2}\). Simplify the right - hand side: \(x = 2\sqrt{2}\).

Answer:

\(x = 2\sqrt{2}\)