Question

find cos(θ) for the following triangle. a right triangle has vertices q, p, r with a right angle at p. side pq is labeled 5 cm, side pr is labeled 12 cm, and angle θ is at q

Explanation:

Step1: Find the hypotenuse using Pythagorean theorem

In a right - triangle, by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 5\), \(b = 12\). Then \(c=\sqrt{5^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\)

Step2: Use the cosine formula

The cosine of an angle \(\theta\) in a right - triangle is given by \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For angle \(\theta\), the adjacent side is \(PQ = 5\) and the hypotenuse is \(13\). So \(\cos\theta=\frac{5}{13}\)

Answer:

\(\frac{5}{13}\) (corresponds to the orange - colored option)