AMC 10 · 2010 · #20

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
tangent-circlessimilar-figuresarea-circles identify-subproblems ↑ Prerequisites: area-circles
📏 Medium solution 💡 3 insights
Problem
A regular hexagon ABCDEF has two circles lying outside it. Both circles are tangent to the full lines BC and FA. The first circle is also tangent to side AB, and the second circle is also tangent to side DE. Find the ratio of the area of the second circle to the area of the first circle.

Pick an answer.

(A)
18
(B)
27
(C)
36
(D)
81
(E)
108

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

This is a pure shapes-and-positions problem, so Tool #1 (Draw a Diagram) is the anchor: extending lines BC and FA until they cross reveals a 60° corner that both circles must live in. Tool #7 (Identify Subproblems) then splits the work into one reusable fact — where a circle sits inside that corner — plus two separate radius calculations, one from AB and one from DE. Tool #4 (Introduce a Variable) fixes the hexagon side length at 1 so every distance becomes a concrete number and the two radii can be compared.

1STEP 1

Extend the two shared lines to a corner

Extend lines BC and FA to meet at P. Since the interior angles are 120°, that corner is 60°, and both circles center on its bisector.

∠ between lines BC and FA = 180° - 120° = 60°
2STEP 2

A circle in the corner sits twice its radius from the tip

Bisecting the corner makes a 30-60-90 triangle whose short leg is the radius r, so the tip-to-center distance is 2r for either circle.

distance from P to center = 2r
3STEP 3

First circle: measure from AB

Set the side to 1. Then P, B, A make an equilateral triangle of height √(3)/2, and that height is 2r₁ + r₁, so r₁ = √(3)/6.

3r₁ = √(3)/2 → r₁ = √(3)/6
4STEP 4

Second circle: measure from DE, farther out

This circle pokes past DE, so P-to-DE is only r₂; that same span is √(3)/2 down to AB plus the hexagon width √(3), giving r₂ = 3√(3)/2.

r₂ = √(3)/2 + √(3) = 3√(3)/2
5STEP 5

Square the radius ratio

Area scales as the square of the radius, and r₂/r₁ = 9, so the area ratio is 9² = 81, choice (D).

(π r₂²)/(π r₁²) = (r₂/r₁)² = 9² = 81
Answer
81
The radii are r₁ = √(3)/6 ≈ 0.29 and r₂ = 3√(3)/2 ≈ 2.60, a ratio of about 9 — the second circle really is much larger, which matches a picture where it wraps the far side of the hexagon. Squaring 9 gives 81, one of the choices. The subtle point that keeps the answer from being 9 (the linear factor) is that areas scale as the square, and the answer being 81 = 9² rather than, say, 3² = 9 comes from the near circle tucking in before AB (three radii from the tip) while the far circle pokes out past DE (one radius from the tip). Answer (D) holds.
💡Key takeaway

Extend the two shared lines to a corner, remember a circle's center sits two radii from the tip, then square the radius ratio to get the area ratio.

  • Extend the two shared lines to a corner
  • A circle in the corner sits twice its radius from the tip
  • First circle: measure from AB
  • Second circle: measure from DE, farther out
  • Square the radius ratio

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