Problem 13 · 2014 Math Kangaroo
Medium
Geometry & Measurement
symmetry
The ratio of the radii of two concentric circles is 1 : 3. The line AC is a diameter of the bigger circle. A chord BC of the big circle touches the small circle (see diagram). The line AB has length 12. How big is the radius of the big circle?
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Answer: B — 18
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Hint 1 of 2
Because AC is a diameter, the angle at B is a right angle.
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Hint 2 of 2
The centre is the midpoint of AC; how far is a midpoint of the hypotenuse from one leg?
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Approach: use the right angle at B and a midline distance
- AC is a diameter, so by Thales the angle at B is 90°; thus AB ⊥ BC.
- Chord BC is tangent to the small circle, so the centre O lies a distance r = R/3 from line BC.
- O is the midpoint of AC, and its distance to line BC is half of AB (a midline), i.e. 12/2 = 6.
- Hence R/3 = 6, giving R = 18.
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