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 triangle parts

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

Step2: Apply Law of Sines to find $\theta$

Relate $\angle R$, $QS$, $\theta$, $QR$:
$$\frac{\sin(80^\circ)}{3.1} = \frac{\sin(\theta)}{2.4}$$
Solve for $\sin(\theta)$:
$$\sin(\theta) = \frac{2.4 \times \sin(80^\circ)}{3.1}$$
Calculate $\sin(80^\circ) \approx 0.9848$:
$$\sin(\theta) \approx \frac{2.4 \times 0.9848}{3.1} \approx 0.767$$
Find $\theta \approx \arcsin(0.767) \approx 50^\circ$

Step3: Calculate $\angle Q$

Sum of angles in triangle is $180^\circ$:
$$\angle Q = 180^\circ - 80^\circ - 50^\circ = 50^\circ$$

Step4: Apply Law of Sines to find $RS$

Relate $\angle Q$, $RS$, $\angle S$, $QR$:
$$\frac{\sin(50^\circ)}{x} = \frac{\sin(50^\circ)}{2.4}$$
Since $\angle Q = \angle S$, $x = 2.4$ (or solve explicitly:
$$x = \frac{2.4 \times \sin(50^\circ)}{\sin(50^\circ)} = 2.4$$
)

Answer:

2.4 units