Problem 11 · 2017 Math Kangaroo
Hard
Geometry & Measurement
pythagorean-triple
Three circles with centres A, B, C touch each other in pairs from the outside (see diagram). Their radii are 3, 2 and 1. How big is the area of the triangle ABC?
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Answer: A — 6
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Hint 1 of 2
When two circles touch externally, the distance between their centres is the sum of the radii.
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Hint 2 of 2
Work out the three side lengths of triangle ABC; they may form a familiar right triangle.
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Approach: get the side lengths from touching circles, recognise a 3-4-5 triangle
- Touching externally, AB = 3 + 2 = 5, BC = 2 + 1 = 3, CA = 1 + 3 = 4.
- Sides 3, 4, 5 form a right triangle (3^2 + 4^2 = 5^2).
- Its area is (3 x 4)/2 = 6.
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