Question

d. c =
8\
65°
c
480
x

Explanation:

Step1: Identify triangle type

The triangle is right - angled (one right angle, \(90^{\circ}\)) and has a \(65^{\circ}\) angle.

Step2: Use triangle angle sum

The sum of angles in a triangle is \(180^{\circ}\). Let the right angle be \(90^{\circ}\), one angle be \(65^{\circ}\), and angle \(C\) be the hypotenuse - related angle. Wait, actually, for a right - angled triangle, to find the length of the hypotenuse \(C\) (assuming the side of length \(8\) is adjacent to the \(65^{\circ}\) angle), we use the cosine function. The cosine of an angle in a right - angled triangle is defined as \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 65^{\circ}\), adjacent side \(a = 8\) inches, and hypotenuse \(C\). So \(\cos(65^{\circ})=\frac{8}{C}\), then \(C=\frac{8}{\cos(65^{\circ})}\).

Step3: Calculate \(\cos(65^{\circ})\)

We know that \(\cos(65^{\circ})\approx0.4226\).

Step4: Compute \(C\)

\(C=\frac{8}{0.4226}\approx18.93\) inches (rounded to two decimal places).

Wait, maybe the problem is to find the angle? Wait, no, the label \(C\) is on the hypotenuse. Wait, if it's a right - angled triangle, the sum of the non - right angles is \(90^{\circ}\). Wait, the right angle is \(90^{\circ}\), one angle is \(65^{\circ}\), so the other non - right angle is \(90 - 65=25^{\circ}\)? No, that can't be. Wait, maybe I misread. Wait, the triangle has a right angle (the square corner), a \(65^{\circ}\) angle, and \(C\) is the hypotenuse. So to find the length of \(C\) (the hypotenuse), we use trigonometry. The side of length \(8\) is adjacent to the \(65^{\circ}\) angle. So \(\cos(65^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{8}{C}\), so \(C = \frac{8}{\cos(65^{\circ})}\). Calculating \(\cos(65^{\circ})\approx0.4226\), then \(C=\frac{8}{0.4226}\approx18.93\) inches.

Answer:

\(C\approx18.93\) inches (or if we consider angle, but the label \(C\) is on the side, so it's length. If it's angle, then the other non - right angle is \(25^{\circ}\), but the label \(C\) is on the side. So most probably, it's the length of the hypotenuse, so \(C\approx18.9\) (or more precise \(18.93\)) inches.