Question

consider triangle xyz. what is the approximate length of side yz? a. 15.7 units b. 13.3 units c. 38.2 units d. 47.1 units

Explanation:

Step1: Find the third - angle

The sum of angles in a triangle is 180°. Let the third angle be $\angle Z$. So, $\angle Z=180^{\circ}-(32^{\circ} + 94^{\circ})=54^{\circ}$.

Step2: Apply the Law of Sines

The Law of Sines states that $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. We want to find the length of $YZ$. Let $YZ = a$, $XY = c = 24$, $\angle X = 32^{\circ}$, and $\angle Z=54^{\circ}$. Then $\frac{a}{\sin X}=\frac{c}{\sin Z}$, so $a=\frac{c\sin X}{\sin Z}$.

Step3: Substitute the values

Substitute $c = 24$, $\sin X=\sin32^{\circ}\approx0.5299$, and $\sin Z=\sin54^{\circ}\approx0.8090$ into the formula. $a=\frac{24\times0.5299}{0.8090}=\frac{12.7176}{0.8090}\approx15.7$.

Answer:

A. 15.7 units