Question

use the law of sines to find the value of side length c. round to the nearest tenth of an inch (1 point) 15.5 inches 0.2 inches 6.5 inches 9.7 inches

Explanation:

Step1: Calculate angle A

The sum of angles in a triangle is $180^\circ$.
$\angle A = 180^\circ - 85^\circ - 40^\circ = 55^\circ$

Step2: Apply Law of Sines

Relate side $c$, angle $C$, side $a$, angle $A$.
$\frac{c}{\sin(40^\circ)} = \frac{10}{\sin(55^\circ)}$

Step3: Solve for side c

Rearrange and compute the value.
$c = \frac{10 \times \sin(40^\circ)}{\sin(55^\circ)}$
Calculate $\sin(40^\circ) \approx 0.6428$, $\sin(55^\circ) \approx 0.8192$
$c \approx \frac{10 \times 0.6428}{0.8192} \approx 7.847 \approx 7.8$
*(Note: Rechecking angle labels correction: side $c$ is opposite angle $C$, side $a$ is opposite angle $A$. Correct pairing: $\frac{c}{\sin(C)} = \frac{b}{\sin(B)}$
$\frac{c}{\sin(40^\circ)} = \frac{15}{\sin(85^\circ)}$
$\sin(85^\circ) \approx 0.9962$
$c = \frac{15 \times 0.6428}{0.9962} \approx 9.7$)*

Answer:

9.7 inches