In circle O, minor arc AB is subtended by a central angle of 120^\circ, and the radius of … — GRE General Test

In circle $O$ , minor arc $AB$ is subtended by a central angle of $120^\circ$ , and the radius of circle $O$ is $6$ . A smaller circle, circle $C$ , is constructed such that it is tangent to chord $AB$ at its midpoint and tangent to minor arc $AB$ at its midpoint. What is the area of a sector of circle $C$ subtended by a central angle of $90^\circ$ ?

$\frac{9\pi}{16}$ Answer
B
$\frac{9\pi}{8}$
C
$\frac{9\pi}{4}$
D
$\frac{27\pi}{16}$
E
$9\pi$

Answer

\frac{9\pi}{16}

The distance from the center of circle

O

to chord

AB

6 \cos(60^\circ) = 3

, leaving a height of

6 - 3 = 3

for the minor segment. Circle

C

is tangent to the chord and arc midpoints, making its diameter equal to

3

and its radius

\frac{3}{2}

. The area of a

90^\circ

sector of circle

C

\frac{90^\circ}{360^\circ} \pi \left(\frac{3}{2}\right)^2 = \frac{9\pi}{16}

Step-by-Step Solution

Find the perpendicular distance from center

O

to chord

AB

In isosceles triangle

OAB

with

OA = OB = 6

and central angle

\angle AOB = 120^\circ

, the perpendicular bisector from

O

AB

meets

AB

at midpoint

M

\angle AOM = 60^\circ

, so

OM = OA \cos(60^\circ) = 6 \times \frac{1}{2} = 3

Determining

OM

allows us to find the sagitta (height) of the minor segment bounded by chord

AB

and minor arc

AB

Calculate the height of the minor segment (diameter of circle

C

The height of the minor segment along the radius passing through

M

to arc midpoint

N

MN = ON - OM = 6 - 3 = 3

. Since circle

C

is tangent to

AB

M

and to the arc at

N

, segment

MN

is a diameter of circle

C

, giving a diameter of

3

The space between the chord midpoint and arc midpoint bounds circle

C

, defining its diameter.

Find the radius of circle

C

Radius

r_C = \frac{\text{diameter}}{2} = \frac{3}{2}

The radius of circle

C

is needed to calculate its sector area.

Compute the area of the

90^\circ

sector of circle

C

\text{Area} = \frac{\theta}{360^\circ} \pi r_C^2 = \frac{90^\circ}{360^\circ} \pi \left(\frac{3}{2}\right)^2 = \frac{1}{4} \pi \left(\frac{9}{4}\right) = \frac{9\pi}{16}.

Multiply the fraction of the full circle represented by the central angle by the total area of circle

C

Key Concept

Calculating sector area of an inscribed circle within a circle segment

Estimated Time:2m 0s

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