Question

what is the length of line segment rs? use the law of sines to find the answer. round to the nearest tenth. law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$

Explanation:

Step1: Identify the given values

Let $QR = 2.4$, $\angle R=80^{\circ}$, $QS = 3.1$. We want to find $RS$.

Step2: Apply the law of sines

By the law of sines, $\frac{\sin R}{QS}=\frac{\sin S}{QR}$. First, find $\sin S$.
$\sin S=\frac{QR\times\sin R}{QS}$. Substitute $QR = 2.4$, $\angle R = 80^{\circ}$, $QS = 3.1$.
$\sin S=\frac{2.4\times\sin80^{\circ}}{3.1}$. Since $\sin80^{\circ}\approx0.9848$, then $\sin S=\frac{2.4\times0.9848}{3.1}=\frac{2.36352}{3.1}\approx0.7624$.
So, $S=\sin^{- 1}(0.7624)\approx49.6^{\circ}$.

Step3: Find the third - angle

The sum of angles in a triangle is $180^{\circ}$. Let $\angle Q=180^{\circ}-\angle R-\angle S$.
$\angle Q = 180^{\circ}-80^{\circ}-49.6^{\circ}=50.4^{\circ}$.

Step4: Use the law of sines again to find $RS$

$\frac{RS}{\sin Q}=\frac{QS}{\sin R}$.
$RS=\frac{QS\times\sin Q}{\sin R}$. Substitute $QS = 3.1$, $\sin Q=\sin50.4^{\circ}\approx0.7707$, $\sin R=\sin80^{\circ}\approx0.9848$.
$RS=\frac{3.1\times0.7707}{0.9848}=\frac{2.38917}{0.9848}\approx2.4$.

Answer:

2.4 units