Question

in the diagram below, write the ratio for cos p as a simplified fraction.

Explanation:

Step1: Identify adjacent/hypotenuse for ∠P

In right triangle \(PRQ\), for angle \(P\):

• Adjacent side = \(PR = 8\)
• Opposite side = \(QR = 12\)

First, calculate hypotenuse \(QP\) via Pythagorean theorem:
$$QP = \sqrt{PR^2 + QR^2} = \sqrt{8^2 + 12^2}$$

Step2: Compute hypotenuse length

$$QP = \sqrt{64 + 144} = \sqrt{208} = 4\sqrt{13}$$

Step3: Apply cosine definition

Cosine of an angle is $\frac{\text{adjacent}}{\text{hypotenuse}}$:
$$\cos P = \frac{8}{4\sqrt{13}} = \frac{2}{\sqrt{13}} = \frac{2\sqrt{13}}{13}$$

Answer:

$\frac{2\sqrt{13}}{13}$