AMC 10 · 2004 · #23

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
tangent-circlescoordinate-geometrypythagorean-theorem convert-to-algebrasymmetry-argument ↑ Prerequisites: pythagorean-theorem
📏 Long solution 💡 2 insights 📊 Diagram
Problem
Three circles A, B, C each touch the other two from outside and all sit inside a big circle D, touching it from within. Circles B and C are the same size. Circle A has radius 1 and its edge passes through the center of D. Find the radius of circle B.

Pick an answer.

(A)
$\frac23$
(B)
$\frac {\sqrt3}{2}$
(C)
$\frac78$
(D)
$\frac89$
(E)
$\frac {1 + \sqrt3}{3}$

AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Every fact in this problem is about distances between circle centers, so the plan is to place the centers on a coordinate grid (Tool #1) and call the unknown radius of B a letter r (Tool #4). Tangency turns each 'touch' into a clean distance equation: the distance between two centers equals the sum of radii when they touch outside, or the difference when one is inside the other. First a small subproblem (Tool #7) fixes the size of the big circle D from the clue that A passes through its center. Then two tangency equations for circle B — one against D, one against A — plus the up-down symmetry that pins B's height, give enough equations to solve for r. Subtracting the two circle equations is the key move: the squared terms cancel and leave a simple line, which feeds straight back to a single equation in r.

1STEP 1

Find the big circle's radius

A's edge reaches D's center, so A's diameter stretches from there to D's rim: big circle D has radius 2.

R_D = 2 × r_A = 2 × 1 = 2
2STEP 2

Set up coordinates and name the radius

Put D at (0,0) and A at (-1,0). Let B have radius r and center (x,y); C is B's mirror image, center (x,-y).

D=(0,0), R_D=2; A=(-1,0), r_A=1; B=(x,y), r_B=r
3STEP 3

Write the two tangency equations for B

B touches D from inside, so its distance to (0,0) is 2-r; B touches A from outside, so its distance to (-1,0) is 1+r.

x²+y²=(2-r)² (x+1)²+y²=(1+r)²
4STEP 4

Subtract to get x, and use symmetry for y

Subtracting the two squared equations kills x² and y² and leaves x = 3r-2; B touching C gives 2y = 2r, so y = r.

2x+1 = 6r-3 → x = 3r-2; 2y = 2r → y = r
5STEP 5

Substitute and solve for r

Feeding those into x²+y²=(2-r)² collapses to 9r²-8r=0, and a radius cannot be 0, so r = 8/9, choice (D).

(3r-2)²+r²=(2-r)² → 9r²-8r=0 → r=8/9
Answer
8/9
Check the radius r=8/9 against the picture. Then x=3r-2=2/3 and y=r=8/9, so B's center is (2/3,8/9). Distance to D's center: √((2/3)²+(8/9)²)=√(100/81)=10/9=2-8/9, matching the inside-touch. Distance to A's center (-1,0): √((5/3)²+(8/9)²)=√(289/81)=17/9=1+8/9, matching the outside-touch. Both tangency conditions hold, so 8/9 is right. It also passes the eye test: B is a bit smaller than A (radius 1), and 8/9 is just under 1. The trap answer (C) 7/8 is close but fails the distance checks; (B) and (E) carry a √3 that never appears once you use the given radius 1.
💡Key takeaway

When circles touch, the distance between their centers is just the sum or difference of the radii — put the centers on a grid and every 'touch' becomes an equation you can solve.

  • Find the big circle's radius
  • Set up coordinates and name the radius
  • Write the two tangency equations for B
  • Subtract to get x, and use symmetry for y
  • Substitute and solve for r

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