Problem 28 · 2022 Math Kangaroo
Stretch
Geometry & Measurement
sum-constraintcasework
Consider the five circles with midpoints A, B, C, D and E respectively, which touch each other as displayed in the diagram. The line segments, drawn in, connect the midpoints of adjacent circles. The distances between the midpoints are AB = 16, BC = 14, CD = 17, DE = 13 and AE = 14. Which of the points is the midpoint of the circle with the biggest radius?
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Answer: A — A
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Hint 1 of 2
Each connecting segment length equals the sum of the two touching circles' radii.
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Hint 2 of 2
Set up r_A+r_B = 16 and the rest, then solve for the radii around the ring.
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Approach: turn each touching pair into a radius-sum equation and solve
- Touching circles meet where their radii add, so \(r_A+r_B=16\), \(r_B+r_C=14\), \(r_C+r_D=17\), \(r_D+r_E=13\), \(r_E+r_A=14\).
- Subtracting pairs gives \(r_A-r_C=2\) and \(r_C-r_E=4\); putting these into \(r_E+r_A=14\) yields \(r_C=8\).
- Then \(r_A=10, r_B=6, r_D=9, r_E=4\), so the largest radius is at point A.
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