Exercise 54 Page 656

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. For simplicity, let's label the endpoints of the involved arcs.

Looking at the diagram, we notice that the central angle of the minor arc AB is a right angle. Recall that the measure of a minor arc is equal to the measure of its corresponding central angle. Therefore, m AB= 90.

We can also notice that the minor arcs AB and BC are adjacent arcs and form the semicircle ABC with an arc measure of 180^(∘). 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. m AB+m BC=180^(∘) Let's substitute 90 for m AB in the above equation and solve for m BC.

mAB+mBC=180

Substitute

mAB= 90

90+mBC=180

SubEqn

LHS-90=RHS-90

mBC=90

Therefore, the measure of the arc BC is 90^(∘).

We can also see that the radius of the circle is 6 meters. Let's substitute m BC= 90 and r=6 in the formula for arc length and simplify.

Length of BC=mBC/360 * 2π r

SubstituteII

mBC= 90, r= 6

Length of BC=90/360 * 2π (6)

▼

Evaluate right-hand side

CommutativePropMult

Commutative Property of Multiplication

Length of BC=90/360 * 2(6)π

Multiply

Multiply

Length of BC=90/360 * (12)π

MoveRightFacToNum

a/c* b = a* b/c

Length of BC=1080/360 * π

CalcQuot

Calculate quotient

Length of BC=3π

The length of the arc is 3π meters.