Question

#22 is what I need help with. I'm not sure how to find the side lengths..

Answer 1

∠C = 68°

Side length AC = 22.9

Side length BC = 14.3

Step-by-step 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:

`(b)/(\sin B)=(c)/(\sin C)`

Substitute c = 22, B = 75° and C = 68° into the sine rule formula above

`\begin{gathered} (b)/(\sin75\degree)=(22)/(\sin 68\degree) \\ (b)/(0.9659)=(22)/(0.9272) \\ \text{Cross multiply} \\ 0.9272b=21.2498 \\ \text{Divide both sides by 0.9272} \\ (0.9272b)/(0.9272)=(21.2498)/(0.9272) \\ b=22.918 \\ To\text{ the nearest tenth,} \\ b=22.9 \end{gathered}`

So side length IACI = 22.9

To find IBCI

Let IBCI = a

Using Sine rule:

`(a)/(\sin A)=(c)/(\sin C)`

Plug in c = 22, A = 37°, and C = 68° into the formula

`\begin{gathered} (a)/(\sin37\degree)=(22)/(\sin 68\degree) \\ (a)/(0.6018)=(22)/(0.9272) \\ \text{Cross multiply} \\ 0.9272a=22*0.6018 \\ \text{0}.9272a=13.2396 \\ \text{Divide both sides by 0.9272} \\ (0.9272a)/(0.9272)=(13.2396)/(0.9272) \\ a=14.279 \\ To\text{ the nearest tenth,} \\ a=14.3 \end{gathered}`

Therefore, side length IBCI = 14.3