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: $\frac{sin(a)}{a} = \frac{sin(b)}{b} = \frac{sin(c)}{c}$
2.2 units
2.4 units
3.0 units
3.3 units

Explanation:

Step1: Define variables for Law of Sines

Let $\angle R = 80^\circ$, $QS = 3.1$, $QR = 2.4$, $\angle S = x$, and $RS = y$.

Step2: Find $\angle S$ using Law of Sines

Set up ratio for $\angle R$, $QS$, $\angle S$, $QR$:
$$\frac{\sin(80^\circ)}{3.1} = \frac{\sin(x)}{2.4}$$
Solve for $\sin(x)$:
$$\sin(x) = \frac{2.4 \times \sin(80^\circ)}{3.1}$$
Calculate $\sin(80^\circ) \approx 0.9848$:
$$\sin(x) \approx \frac{2.4 \times 0.9848}{3.1} \approx \frac{2.3635}{3.1} \approx 0.7624$$
Find $x$:
$$x \approx \arcsin(0.7624) \approx 49.6^\circ$$

Step3: Calculate $\angle Q$

Sum of angles in triangle is $180^\circ$:
$$\angle Q = 180^\circ - 80^\circ - 49.6^\circ = 50.4^\circ$$

Step4: Find $RS$ using Law of Sines

Set up ratio for $\angle Q$, $RS$, $\angle R$, $QS$:
$$\frac{\sin(50.4^\circ)}{y} = \frac{\sin(80^\circ)}{3.1}$$
Solve for $y$:
$$y = \frac{3.1 \times \sin(50.4^\circ)}{\sin(80^\circ)}$$
Calculate $\sin(50.4^\circ) \approx 0.7705$:
$$y \approx \frac{3.1 \times 0.7705}{0.9848} \approx \frac{2.3886}{0.9848} \approx 2.4$$

Answer:

2.4 units