Question

Suppose that circles R and S have a central angle measuring 80°. Additionally, the measure of the sector for circle R is

32
9
π m2 and for circle S is 18π m2.

If the radius of circle R is 4 m, what is the radius of circle S?
A) 6 m
B) 9 m
C) 12 m
D) 15 m

Answer 1

B) 9 m

Explanation:

32

9

π

18π

=

42

x2

x = 9

When circles have the same central angle measure, the ratio of measure of the sectors is the same as the ratio of the radii squared.

Answer 2

The correct answer is B) 9 m.

The measure of the sector of circle R is 32π/9 m. The measure of the central angle is 80°. This means that the sector is 80/360 = 2/9 of the circle. The area of a circle is given by A=πr², so the area of the sector is A=2/9πr². To verify this, 2/9π(4²) = 2/9π(16) = 32π/9.

Using this same formula for circle S, we will work backward to find the radius:

18π = 2/9πr²

Multiply both sides by 9:
18*9π = 2πr²
162π = 2πr²

Divide both sides by 2π:
162π/2π = 2πr²/2π
81 = r²

Take the square root of both sides:
√81 = √r²
9 = r