Exercise 47 Page 677

Let's begin with recalling the Law of Sines. If △ ABC has lengths of a, b, and c and angle measures of A, B, and C, then we can write that the ratios of the sine of an angle to the side opposite this angle are equal.

In our exercise we are given a triangle and asked to evaluate its perimeter. Let's take a look at the given picture and name vertices with consecutive letters.

First, we can evaluate the measure of ∠ A. To do this, let's use the Triangle Angle Sum Theorem.

m∠ A+ 31^(∘)+ 86^(∘)=180^(∘)

AddTerms

Add terms

m∠ A+117^(∘)=180^(∘)

SubEqn

LHS-117^(∘)=RHS-117^(∘)

m∠ A=63^(∘)

The third angle has a measure of 63^(∘). Let's add this information to our diagram. Let a and b represent the lengths of AB and AC.

Since we are asked to evaluate the perimeter, we need to know all the side lengths. To do this, we will create a proportion using the Law of Sines. sin 31^(∘)/a=sin 63^(∘)/9=sin 86^(∘)/b Let's start with evaluating the value of a using cross multiplication.

sin 31^(∘)/a=sin 63^(∘)/9

CrossMult

Cross multiply

9sin31^(∘)=asin63^(∘)

DivEqn

.LHS /sin63^(∘).=.RHS /sin63^(∘).

9sin31^(∘)/sin63^(∘)=a

RearrangeEqn

Rearrange equation

a=9sin31^(∘)/sin63^(∘)

UseCalc

Use a calculator

a=5.2023...

RoundDec

Round to 1 decimal place(s)

a≈ 5.2

Now we will solve for b.

sin 63^(∘)/9=sin 86^(∘)/b

CrossMult

Cross multiply

bsin63^(∘)=9sin86^(∘)

DivEqn

.LHS /sin63^(∘).=.RHS /sin63^(∘).

b=9sin86^(∘)/sin63^(∘)

UseCalc

Use a calculator

b=10.0763...

RoundDec

Round to 1 decimal place(s)

b≈ 10.1

Let's add this information to our diagram.

Now we can evaluate the perimeter of the triangle by adding all side lengths. Remember that this will be an approximation as we are using approximate side lengths. 5.2+ 10.1+ 9=24.3 The perimeter is approximately 24.3 units.