Question

find the length of side c.
round to the nearest tenth.

Explanation:

Step1: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know \(A = 23^{\circ}\), \(B=32^{\circ}\), \(C = 125^{\circ}\), and \(b = 9.5\). We want to find \(c\), so we use \(\frac{c}{\sin C}=\frac{b}{\sin B}\).

Step2: Substitute the values

Substitute \(b = 9.5\), \(B = 32^{\circ}\), and \(C=125^{\circ}\) into the formula: \(c=\frac{b\times\sin C}{\sin B}\).

$$c=\frac{9.5\times\sin(125^{\circ})}{\sin(32^{\circ})}$$

Step3: Calculate the sine values

We know that \(\sin(125^{\circ})\approx0.8192\) and \(\sin(32^{\circ})\approx0.5299\).

$$c=\frac{9.5\times0.8192}{0.5299}$$

Step4: Perform the calculation

First, calculate \(9.5\times0.8192 = 7.7824\). Then, \(c=\frac{7.7824}{0.5299}\approx14.7\)

Answer:

\(14.7\)