AMC 10 · 2018 · #15

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
tangent-circlessimilar-trianglesratio-proportion identify-subproblems ↑ Prerequisites: similar-trianglestangent-circles
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
A big circle has radius 13. Two equal circles of radius 5 sit inside it. The two small circles touch each other, and each one touches the big circle from the inside at a point (A and B). Find the distance AB, write it as a reduced fraction m/n, and report m+n.

Pick an answer.

(A)
21
(B)
29
(C)
58
(D)
69
(E)
93

AMC 10 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

Tangency is about which points lie on which lines, so a clean labeled diagram is the key. Mark the big center O and the two small centers. Two facts about tangent circles turn the picture into two triangles that share the angle at O, and those triangles are similar. The whole problem then becomes one proportion.

1STEP 1

Place the centers

Tangency puts each touch point on the line through both centers, so A sits on ray OC1 and B sits on ray OC2.

A on ray OC₁, B on ray OC₂
2STEP 2

Find the key distances

Internal tangency gives OC1 = OC2 = 13 - 5 = 8, and external tangency gives C1C2 = 5 + 5 = 10.

OC₁ = OC₂ = 13 - 5 = 8, C₁C₂ = 5 + 5 = 10
3STEP 3

Spot the similar triangles

Triangles OC1C2 and OAB share the angle at O, and OA/OC1 = OB/OC2 = 13/8, so by SAS they are similar.

△ OAB ∼ △ OC₁C₂, OA/OC₁ = OB/OC₂ = 13/8
4STEP 4

Set up the proportion for AB

Since the scale factor is 13/8, AB = 13/8 · C1C2 = 13/8 · 10 = 65/4.

AB = 13/8· C₁C₂ = 13/8· 10 = 130/8 = 65/4
5STEP 5

Reduce and add

65 and 4 share no common factor, so m = 65, n = 4 and m + n = 65 + 4 = 69, choice (D).

gcd(65,4)=1, m+n = 65 + 4 = 69
Answer
69
AB = 65/4 = 16.25 is a bit longer than the big circle's radius 13 but well under the diameter 26, which fits two tangent points near the bottom that are spread fairly wide apart. The answer choices 21, 29, 58, 69, 93 are all m+n values; only the reduced fraction 65/4 gives a member of the list, 69, so the match confirms the work.
💡Key takeaway

Tangent points line up with the centers, so the small triangle and big triangle are scaled copies, and the answer is just that scale factor times the easy distance.

  • Place the centers
  • Find the key distances
  • Spot the similar triangles
  • Set up the proportion for AB
  • Reduce and add

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