AMC 10 · 2007 · #18

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length
tangent-circlespythagorean-theoremsymmetry-argument convert-to-algebra ↑ Prerequisites: pythagorean-theorem
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
A unit circle (radius 1) is ringed by four equal circles of radius r. Each outer circle touches the central circle, and each touches its two neighbors. Find r.

Pick an answer.

(A)
$\sqrt{2}$
(B)
$1+\sqrt{2}$
(C)
$\sqrt{6}$
(D)
3
(E)
$2+\sqrt{2}$

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The picture is the whole problem, so first draw the centers (Tool #1) and turn each "touch" into a distance between centers. The symmetry places the four outer centers at the corners of a square (Tool #17), which hands us a right angle at the middle. That right triangle converts the geometry into a Pythagorean equation (Tool #13) with the single unknown r (Tool #4). Solving and matching the exact value to a listed choice finishes it (Tool #3).

1STEP 1

Turn each touch into a distance

Center the unit circle at O. Tangency makes O-to-outer-center 1+r, and two touching neighbors 2r apart.

O to an outer center=1+r, adjacent outer centers=2r
2STEP 2

Read the right angle from symmetry

Evenly spaced centers form a square about O, so neighbors P and Q subtend 360°÷4=90° at O, with OP=OQ=1+r, PQ=2r.

∠ POQ=360°/4=90°, OP=OQ=1+r, PQ=2r
3STEP 3

Apply the Pythagorean theorem

Right-angled at O, triangle OPQ gives OP²+OQ²=PQ²; substituting the two lengths yields 2(1+r)²=4r².

(1+r)²+(1+r)²=(2r)² → 2(1+r)²=4r²
4STEP 4

Simplify to a clean equation

Halve both sides to (1+r)²=2r², take the positive root for 1+r=r√(2), then collect the r terms: 1=r(√(2)-1).

(1+r)²=2r² → 1+r=r√(2) → 1=r(√(2)-1)
5STEP 5

Solve and rationalize

So r=1/(√(2)-1); multiplying top and bottom by √(2)+1 makes the denominator 1, leaving r=1+√(2), choice (B).

r=1/(√(2)-1)=(√(2)+1)/(√(2)-1)(√(2)+1)=(√(2)+1)/1=1+√(2)
Answer
1+√(2)
Check r=1+√(2)≈2.414. Then O to an outer center is 1+r≈3.414 and neighboring centers are 2r≈4.828 apart. Test the right triangle: 2·(3.414)²≈23.3 and (4.828)²≈23.3 — they match. The outer circles must be bigger than the unit circle, and 2.414 ＞ 1 fits. The nearby decoy √(6)≈2.449 is close but fails the exact equation, so the geometry pins the value to 1+√(2), choice (B).
💡Key takeaway

When circles are packed so they just touch, the distance between two centers is the sum of their radii — turn every touch into that length and the picture becomes an equation.

  • Turn each touch into a distance
  • Read the right angle from symmetry
  • Apply the Pythagorean theorem
  • Simplify to a clean equation
  • Solve and rationalize

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