Exercise 9 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 ∠ N, MP, and NP one at a time.

Finding m ∠ N

180- 111- 27= 42^(∘)

Finding MP

Now that we know the measure of ∠ N, we can find MP using the Law of Sines. sin N/MP =sin P/MN Let's substitute MN= 50, m ∠ P = 111, and m ∠ N = 42 to isolate MP.

sin N/MP =sin P/MN

SubstituteValues

Substitute values

sin 42/MP = sin 111/50

▼

Solve for MP

CrossMult

Cross multiply

sin 42 * 50 = MP * sin 111

DivEqn

.LHS /sin 111.=.RHS /sin 111.

sin 42 * 50/sin 111=MP

RearrangeEqn

Rearrange equation

MP=sin 42 * 50/sin 111

UseCalc

Use a calculator

MP=35.836794...

RoundDec

Round to 1 decimal place(s)

MP ≈ 35.8

Finding NP

Finally, we can repeat using the Law of Sines to find NP. sin P/MN =sin M/NP Let's substitute MN= 50, m ∠ P = 111, and m ∠ M = 27 to isolate NP.

sin P/MN =sin M/NP

SubstituteValues

Substitute values

sin 111/50 = sin 27/NP

▼

Solve for NP

CrossMult

Cross multiply

sin 111 * NP = 50 * sin 27

DivEqn

.LHS /sin 111.=.RHS /sin 111.

NP=sin 27 * 50/sin 111

UseCalc

Use a calculator

NP=24.314482...

RoundDec

Round to 1 decimal place(s)

NP ≈ 24.3

Completing the Triangle

With all these calculated, we can complete our diagram.