Exercise 53 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 BC=mBC/360 * π d 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 m AB and m BC form the semicircle ABC, which measures 180^(∘). Therefore, 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 50 for AB in the above equation to find the measure of the arc BC

mAB+mBC=180

Substitute

mAB= 50

50+mBC=180

SubEqn

LHS-50=RHS-50

mBC=130

Therefore, the measure of the arc BC is 130^(∘).

Now we will find the arc length of BC. In the diagram, we can see that the diameter of the circle is 7.2 inches. Let's substitute mBC= 130 and d=7.2 in the formula for arc length and simplify.

Length of BC=mBC/360 * π d

SubstituteII

mBC= 130, d= 7.2

Length of BC=130/360 * π (7.2)

CommutativePropMult

Commutative Property of Multiplication

Length of BC=130/360 * (7.2)π

MoveRightFacToNum

a/c* b = a* b/c

Length of BC=936/360 * π

CalcQuot

Calculate quotient

Length of BC=2.6π

The length of the arc is 2.6π inches.