Question

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

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Here, we know angle $A = 23^{\circ}$, angle $C=125^{\circ}$, side opposite angle $A$ is $b$, and side opposite angle $B$ (where $B = 180^{\circ}-(23^{\circ}+ 125^{\circ})=32^{\circ}$) has length $7$. So, $\frac{b}{\sin125^{\circ}}=\frac{7}{\sin23^{\circ}}$.

Step2: Solve for \(b\)

Cross - multiply: $b=\frac{7\times\sin125^{\circ}}{\sin23^{\circ}}$.
We know that $\sin125^{\circ}=\sin(180^{\circ} - 55^{\circ})=\sin55^{\circ}\approx0.8192$ and $\sin23^{\circ}\approx0.3907$.
Then $b=\frac{7\times0.8192}{0.3907}=\frac{5.7344}{0.3907}\approx14.7$.

Answer:

$14.7$