Answer
∠C = 68°
Side length AC = 22.9
Side length BC = 14.3
Explanation
Given:
∠A = 37°
∠B = 75°
IABI = c = 22
What to find:
∠C, IACI, and IBCI
Step-by-step solution:
To find ∠C
∠A +∠B + ∠C = 180° (sum of angles in a triangle)
37° + 75° + ∠C = 180°
112° + ∠C = 180°
Combine the like terms
∠C = 180° -112°
∠C = 68°
To find IACI
Let IACI = b
Using Sine rule:
[tex]\frac{b}{\sin B}=\frac{c}{\sin C}[/tex]
Substitute c = 22, B = 75° and C = 68° into the sine rule formula above
[tex]\begin{gathered} \frac{b}{\sin75\degree}=\frac{22}{\sin 68\degree} \\ \frac{b}{0.9659}=\frac{22}{0.9272} \\ \text{Cross multiply} \\ 0.9272b=21.2498 \\ \text{Divide both sides by 0.9272} \\ \frac{0.9272b}{0.9272}=\frac{21.2498}{0.9272} \\ b=22.918 \\ To\text{ the nearest tenth,} \\ b=22.9 \end{gathered}[/tex]
So side length IACI = 22.9
To find IBCI
Let IBCI = a
Using Sine rule:
[tex]\frac{a}{\sin A}=\frac{c}{\sin C}[/tex]
Plug in c = 22, A = 37°, and C = 68° into the formula
[tex]\begin{gathered} \frac{a}{\sin37\degree}=\frac{22}{\sin 68\degree} \\ \frac{a}{0.6018}=\frac{22}{0.9272} \\ \text{Cross multiply} \\ 0.9272a=22\times0.6018 \\ \text{0}.9272a=13.2396 \\ \text{Divide both sides by 0.9272} \\ \frac{0.9272a}{0.9272}=\frac{13.2396}{0.9272} \\ a=14.279 \\ To\text{ the nearest tenth,} \\ a=14.3 \end{gathered}[/tex]
Therefore, side length IBCI = 14.3
