Question

find the length of side a.
b
a 50° 35° c
13
b = 95° a = ? c =

Explanation:

Step1: Identify Law of Sines

In a triangle, the Law of Sines states $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Here, side $b = 13$ (opposite angle $B = 95^\circ$), angle $A = 50^\circ$, angle $C = 35^\circ$, and we need to find $a$ (opposite angle $A$? Wait, no: side $a$ is opposite angle $A$? Wait, no, in standard notation, side $a$ is opposite angle $A$, side $b$ opposite angle $B$, side $c$ opposite angle $C$. Wait, the triangle has angle $A = 50^\circ$, angle $C = 35^\circ$, side $b$ (AC) is 13, which is opposite angle $B$. So angle $B = 180 - 50 - 35 = 95^\circ$, which matches. So we need to find side $a$, which is opposite angle $A$? Wait, no: side $a$ is opposite angle $A$? Wait, in the diagram, side $a$ is from $B$ to $C$, so opposite angle $A$. Wait, angle $A$ is at vertex $A$, so side $a$ is $BC$, opposite angle $A$ (50°), side $c$ is $AB$, opposite angle $C$ (35°), and side $b$ is $AC = 13$, opposite angle $B$ (95°). So using Law of Sines: $\frac{a}{\sin A}=\frac{b}{\sin B}$.

Step2: Plug in values

We have $A = 50^\circ$, $B = 95^\circ$, $b = 13$. So $\frac{a}{\sin 50^\circ}=\frac{13}{\sin 95^\circ}$. Solve for $a$: $a=\frac{13 \times \sin 50^\circ}{\sin 95^\circ}$.

Step3: Calculate sines

$\sin 50^\circ \approx 0.7660$, $\sin 95^\circ \approx 0.9962$.

Step4: Compute $a$

$a=\frac{13 \times 0.7660}{0.9962} \approx \frac{9.958}{0.9962} \approx 10.0$.

Answer:

$10.0$ (rounded to the nearest tenth, assuming the problem expects that; if more precision, but likely 10.0)