Problem 6 · 2017 Math Kangaroo
Medium
Geometry & Measurement
area-fraction
The diagram shows a circle with centre O and the diameters AB and CX. Given that OB = BC, which fraction of the circle's area is shaded?
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Answer: B — 13
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Hint 1 of 2
OB is a radius; the condition OB = BC makes triangle OBC special.
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Hint 2 of 2
Find the central angle COB, then the shaded part is a simple fraction of the full 360 degrees.
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Approach: use the equilateral triangle to get the central angle, then the area fraction
- OB and OC are both radii, and OB = BC, so triangle OBC is equilateral and angle COB = 60 degrees.
- By symmetry of the two diameters, the shaded sectors together span 120 degrees of the circle.
- That is 120/360 = 1/3 of the circle's area.
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