Exercise 36 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∠ W, and then the measures of the missing side lengths one at a time.

Finding m ∠ W

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

m∠ W+ 33 + 124 = 180 ⇕ m ∠ W = 23

Finding WS

Note that we know the m ∠ W and the length of the side which is opposite to this angle. We want to find the length of the side, t, that is opposite to the ∠ T. Therefore, we can use the Law of Sines! sin W/w =sin T/t Let's substitute w= 48, m ∠ W= 23, and m ∠ T= 124 to isolate t.

sin W/w =sin T/t

SubstituteValues

Substitute values

sin 23/48 = sin 124/t

▼

Solve for t

CrossMult

Cross multiply

t sin 23 = 48sin 124

DivEqn

.LHS /sin 23.=.RHS /sin 23.

t = 48 sin 124/sin 23

UseCalc

Use a calculator

t = 101.844466...

RoundDec

Round to 1 decimal place(s)

t ≈ 101.8

We found that the length of the side WS is about 101.8 units.

Finding WT

Finally, to find WT, we will use the Law of Sines one more time. sin W/w =sin S/s Let's substitute w= 48, m ∠ W= 23, and m ∠ S= 33 to isolate s.

sin W/w =sin S/s

SubstituteValues

Substitute values

sin 23/48 = sin 33/s

▼

Solve for s

CrossMult

Cross multiply

s sin 23 = 48 sin 33

DivEqn

.LHS /sin 23.=.RHS /sin 23.

s = 48 sin 33/sin 23

UseCalc

Use a calculator

s = 66.907066 ...

RoundDec

Round to 1 decimal place(s)

s ≈ 66.9

We have found the length of the side WT as about 66.9 units.

Completing the Triangle

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