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Question

Difficulty: Very hardCircle Geometry

In a circle with center OO, chord ABAB has length 12312\sqrt{3}. A radius OCOC is perpendicular to chord ABAB and intersects ABAB at point DD. If CD=6CD = 6, what is the area of the sector of the circle bounded by radii OAOA, OBOB, and the minor arc ABAB?

  1. A
    18π18\pi
  2. B
    24π24\pi
  3. 48π48\piAnswer
  4. D
    144π144\pi

Answer

The correct answer is the option representing 48π48\pi.
The correct answer is 48π48\pi because the radius of the circle is determined to be 1212 using the Pythagorean theorem, and the central angle of the sector is 120∘120^\circ. The area is then 120360×π(12)2=48π\frac{120}{360} \times \pi (12)^2 = 48\pi.

Step-by-Step Solution

1
Determine the length of the segment from the midpoint of the chord to its endpoints.
AD=63AD = 6\sqrt{3}
A radius perpendicular to a chord bisects the chord, so DD is the midpoint of chord ABAB.
2
Express the distance from the center OO to the chord intersection DD in terms of the radius RR and set up the Pythagorean theorem for right triangle △ODA\triangle ODA.
R2=(R−6)2+(63)2R^2 = (R - 6)^2 + (6\sqrt{3})^2
The radius OCOC has length RR, making OD=OC−CD=R−6OD = OC - CD = R - 6. Since △ODA\triangle ODA is a right triangle, we can apply the Pythagorean theorem.
3
Solve the equation for the radius RR.
R=12R = 12
Expanding the equation gives R2=R2−12R+36+108R^2 = R^2 - 12R + 36 + 108, which simplifies to 12R=14412R = 144.
4
Find the central angle ∠AOB\angle AOB.
∠AOB=120∘\angle AOB = 120^\circ
In right triangle △ODA\triangle ODA, the cosine of ∠AOD\angle AOD is ODOA=612=12\frac{OD}{OA} = \frac{6}{12} = \frac{1}{2}, which means ∠AOD=60∘\angle AOD = 60^\circ. The total central angle is ∠AOB=2×∠AOD=120∘\angle AOB = 2 \times \angle AOD = 120^\circ.
5
Calculate the area of the sector bounded by OAOA, OBOB, and the minor arc ABAB.
48π48\pi
The area of the sector is the fraction of the circle's total area corresponding to the central angle: 120360×π(12)2=48π\frac{120}{360} \times \pi (12)^2 = 48\pi.

Key Concept

Using perpendicular bisector chord properties and right triangle trigonometry to determine circle sector area
Estimated Time:3m 0s
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Topics

Circle Geometry

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