Exercise 24 Page 543

Let's consider the given diagram.

We are asked to find the value of x and then find the measure of the red arc. We will do it one at a time.

The Value of x

To find the value of x we will write an equation involving x. Note that the minor arcs QR, RS, ST, and TR form a circle. Therefore, the sum of their measures is 360^(∘). mQR+mRS+mST+mTR=360^(∘) On the diagram the measures of all four arcs are expressed in terms of x. These measures are, as follows, mQR= 4x^(∘), mRS= 6x^(∘), mST= 7x^(∘), and mTR= 7x^(∘). Let's substitute these values into the above equation and solve it for x^(∘).

mQR+mRS+mST+mTR=360^(∘)

SubstituteValues

Substitute values

4x^(∘)+ 6x^(∘)+ 7x^(∘)+ 7x^(∘)=360^(∘)

AddTerms

Add terms

24x^(∘)=360^(∘)

DivEqn

.LHS /24.=.RHS /24.

x^(∘)=15^(∘)

Since x^(∘) is equal to 15^(∘), x is equal to 15.

The Measure of the Red Arc

Let's take a look at the given diagram.

We can see that the red arc is a major arc RST. It consists of two adjacent arcs RS and ST whose measures are 6x^(∘) and 7x^(∘), respectively. According to the Arc Addition Postulate the measure of RST is equal to the sum of measures of RS and ST. lThe measure of the red arc =mRST= 6x^(∘)+ 7x^(∘) We have determined that x= 15, so let's substitute 15 for x into the expression for the measure of the red arc.

mRST=6x^(∘)+7x^(∘)

Substitute

x= 15

mRST=6( 15)^(∘)+7( 15)^(∘)

Multiply

Multiply

mRST=90^(∘)+105^(∘)

AddTerms

Add terms

mRST=195^(∘)

The measure of the red arc RST is 195^(∘).