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2016 Math Kangaroo

Problem 27

Problem 27 · 2016 Math Kangaroo Stretch
Geometry & Measurement sum-constraint

The diagram shows a pentagon with the length of each side marked. Five circles are drawn with centres A, B, C, D and E. On each side of the pentagon, the two circles centred at the ends of that side touch each other. Which point is the centre of the biggest circle?

Figure for Math Kangaroo 2016 Problem 27
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Answer: A — A
Show hints
Hint 1 of 3
Touching circles mean neighbouring radii add to the side between their centres.
Still stuck? Show hint 2 →
Hint 2 of 3
Write r_A + r_B = 16, r_B + r_C = 14, r_C + r_D = 17, r_D + r_E = 13, r_E + r_A = 14 and solve.
Still stuck? Show hint 3 →
Hint 3 of 3
Alternating sums of the side lengths give each radius; find the largest.
Show solution
Approach: solve the touching-circles radius system
  1. Touching circles mean neighbouring radii add up to their shared side: rA+rB=16, rB+rC=14, rC+rD=17, rD+rE=13, rE+rA=14.
  2. Solving gives rA=10, rB=6, rC=8, rD=9, rE=4.
  3. The largest radius is rA = 10, so the biggest circle is centred at A.
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