Exercise 8 Page 674

Let's begin by drawing △ ABC and labeling the length of the side and angles. We will also color code the opposite angles and sides. It will help us use the Law of Sines and Law of Cosines later.

Let's find the measures of ∠ B, AB, and BC one at a time.

Finding m ∠ B

180- 32- 71= 77^(∘)

Finding AB

Now that we know the measure of ∠ B, we can find AB using the Law of Sines. sin C/AB =sin B/AC Let's substitute AC= 8, m ∠ B = 77, and m ∠ C = 71 to isolate AB.

sin C/AB =sin B/AC

SubstituteValues

Substitute values

sin 71/AB = sin 77/8

▼

Solve for AB

CrossMult

Cross multiply

sin 71 * 8 = AB * sin 77

DivEqn

.LHS /sin 77.=.RHS /sin 77.

sin 71 * 8/sin 77=AB

RearrangeEqn

Rearrange equation

AB=sin 71 * 8/sin 77

UseCalc

Use a calculator

AB=7.763116...

RoundDec

Round to 1 decimal place(s)

AB ≈ 7.8

Finding BC

Finally, we can repeat using the Law of Sines to find BC. sin A/BC =sin B/AC Let's substitute AC= 8, m ∠ B = 77, and m ∠ A = 32 to isolate BC.

sin A/BC =sin B/AC

SubstituteValues

Substitute values

sin 32/BC = sin 77/8

▼

Solve for BC

CrossMult

Cross multiply

sin 32 * 8 = BC * sin 77

DivEqn

.LHS /sin 77.=.RHS /sin 77.

sin 32 * 8/sin 77=BC

RearrangeEqn

Rearrange equation

BC=sin 32 * 8/sin 77

UseCalc

Use a calculator

BC=4.350866...

RoundDec

Round to 1 decimal place(s)

BC ≈ 4.4

Completing the Triangle

With all these calculated, we can complete our diagram.