Exercise 31 Page 676

Let's begin by color coding the opposite angles, sides, and the vertices in the given triangle. It will help us use the Law of Sines later.

Let's first find the measure of the third interior angle which is m∠ B, and then the measures of the missing side lengths one at a time.

Finding m ∠ B

To find m ∠ B, we can use the Triangle Angle Sum Theorem. This tells us that the measures of the angles in a triangle add up to 180.

94+ m ∠ B+ 52 = 180 ⇔ m ∠ B = 34

Finding AB

Note that we know the m∠ A and the length of the side which is opposite to this angle. We want to find the length of the side AB that is opposite to ∠ C. Therefore, we can use the Law of Sines! sin A/BC =sin C/AB Let's substitute BC= 12, m ∠ A= 94, and m ∠ C= 52 to isolate AB.

sin A/BC =sin C/AB

SubstituteValues

Substitute values

sin 94/12 = sin 52/AB

▼

Solve for AB

CrossMult

Cross multiply

AB * sin 94 = 12 * sin 52

DivEqn

.LHS /sin 94.=.RHS /sin 94.

AB = 12 * sin 52/sin 94

UseCalc

Use a calculator

AB= 9.479219 ...

RoundDec

Round to 1 decimal place(s)

AB ≈ 9.5

We found that the length of the side AB is about 9.5 m.

Finding AC

Finally, to find AC we will use the Law of Sines one more time! sin A/BC =sin B/AC Let's substitute BC= 12, m ∠ A= 94, and m ∠ B= 34 to isolate AC.

sin A/BC =sin B/AC

SubstituteValues

Substitute values

sin 94/12 = sin 34/AC

▼

Solve for AC

CrossMult

Cross multiply

AC * sin 94 = 12 * sin 34

DivEqn

.LHS /sin 94.=.RHS /sin 94.

AC = 12 * sin 34/sin 94

UseCalc

Use a calculator

AC= 6.726700 ...

RoundDec

Round to 1 decimal place(s)

AC ≈ 6.7

We found that the length of the side AC is about 6.7 m.

Completing the Triangle

With all of the angle measures and side lengths, we can complete our diagram.