Exercise 38 Page 729

The length of an arc of a circle is the product of the measure of the arc divided by 360 and the circumference of the circle. Length of QR=mQR/360 * 2π r Therefore, to find the length of the desired arc we first need to find its measure.

Note that the arcs QR and TQ are adjacent. By the Arc Addition Postulate, we know that the measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs. In this case, the arc formed by these two adjacent arcs is a semicircle, which measures 180^(∘). m QR+mTQ=180 We can see in the diagram that mTQ= 112. Let's substitute this value in the above equation, and solve for mQR.

mQR+mTQ=180

Substitute

mTQ= 112

mQR+ 112=180

SubEqn

LHS-112=RHS-112

mQR=68

We found that QR measures 68^(∘).

We are also told that PS, which is the radius of the circle, measures 4 millimeters. Knowing that, we can substitute mQR= 68 and r= 4 in the formula for arc length and simplify.

Length of QR=mQR/360 * 2π r

SubstituteII

mQR= 68, r= 4

Length of QR=68/360 * 2π ( 4)

▼

Evaluate right-hand side

CommutativePropMult

Commutative Property of Multiplication

Length of QR=68/360 * 2(4)π

Multiply

Multiply

Length of QR=68/360 * 8π

MoveRightFacToNum

a/c* b = a* b/c

Length of QR=544π/360

ReduceFrac

a/b=.a /8./.b /8.

Length of QR=68π/45

Remembering that π is approximately 3.142, we can substitute this number into obtained value and simplify.

Length of QR=68π/45

π ≈ 3.142

Length of QR≈68( 3.142)/45

▼

Simplify

Multiply

Multiply

Length of QR≈213.656/45

CalcQuot

Calculate quotient

Length of QR≈ 4.74791...

RoundDec

Round to 2 decimal place(s)

Length of QR≈ 4.75

The length of the arc is about 4.75 millimeters.