Exercise 32 Page 655

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 AB=mAB/360 * 2π r Therefore, to find the length of the desired arc, we first need to find its measure. Let's label the endpoints of the involved arcs.

In the diagram, we see that the central angle of the minor arc BA is a right angle. Recall that the measure of a minor arc is equal to the measure of its corresponding central angle. Therefore, its corresponding minor arc also measures 90^(∘).

Note that the minor arc BA and major arc AB are adjacent arcs. By the Arc Addition Postulate, the arc formed by these two adjacent arcs is a full circle, which means, it measures 360^(∘). m BA+m AB=360^(∘) Let's substitute 90 for mBA in the above equation to find the measure of m AB.

mBA+mAB=360

Substitute

mBA= 90

90+mAB=360

SubEqn

LHS-90=RHS-90

mAB=270

Therefore, the measure of AB is 270^(∘).

We can also see that the radius of the circle is 18 meters. Now, we can substitute m AB= 270 and r=18 in the formula for arc length and simplify.

Length of AB=mAB/360 * 2π r

SubstituteII

mAB= 270, r= 18

Length of AB=270/360 * 2π (18)

▼

Evaluate right-hand side

CommutativePropMult

Commutative Property of Multiplication

Length of AB=270/360 * 2(18)π

Multiply

Multiply

Length of AB=270/360 * 36π

MoveRightFacToNum

a/c* b = a* b/c

Length of AB=9720/360 * π

CalcQuot

Calculate quotient

Length of AB=27π

The length of the arc is 27π meters.