AMC 10 · 2004 · #21

Grade 8 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Nested Frame / Concentric Difference
circular-sectorarea-circleslinear-equations-one-var convert-to-algebraidentify-subproblems ↑ Prerequisites: area-circles
📏 Long solution 💡 2 insights 📊 Diagram
Problem
Three circles share the same center and have radii 3, 2, and 1, so the picture is one disk of radius 3 divided into an inner disk, a middle ring, and an outer ring. Two distinct lines pass through that center, cutting each of those three parts into two opposite narrow wedges and two opposite wide wedges. In the diagram the shading alternates: the narrow wedges are shaded in the outer ring and in the inner disk, while the wide wedges are shaded in the middle ring. Altogether the shaded region is 8/13 of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: π radians is 180 degrees.)

Pick an answer.

(A)
$\frac{\pi}{8}$
(B)
$\frac{\pi}{7}$
(C)
$\frac{\pi}{6}$
(D)
$\frac{\pi}{5}$
(E)
$\frac{\pi}{4}$

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

How to solve
Strategy Introduce a Variable

The whole shaded area depends on just one thing: the acute angle. So name it θ (Tool #4) and everything else can be written in terms of it. The picture looks busy, so break it into the three rings the circles create (Tool #7); each ring's area is easy, and inside each ring the shaded part is just a fraction set by θ. Reading the diagram carefully (Tool #1) shows the shading flips between rings — narrow wedges in the outer and inner rings, wide wedges in the middle ring — which is exactly what makes the totals interesting. Once the shaded area is a formula in θ, the condition 'shaded is 8/13 of unshaded' becomes one equation to solve (Tool #13).

1STEP 1

Split the picture into three rings

Radii 3, 2, 1 make a disk of area 9π split into three rings: inner π, middle 3π, outer 5π (since 4π-π=3π and 9π-4π=5π).

inner=π, middle=4π-π=3π, outer=9π-4π=5π, total=9π
2STEP 2

Name the angle and measure each wedge

Let θ be the acute angle. Each ring holds two narrow wedges of angle θ and two wide of angle π-θ, so the fractions are θ/π and (π-θ)/π.

narrow fraction=2θ/2π=θ/π, wide fraction=(2(π-θ))/2π=(π-θ)/π
3STEP 3

Add up the shaded area

Outer and inner shade the narrow wedges (5θ and θ); the middle shades the wide ones, 3(π-θ). Total shaded 3π+3θ, unshaded 6π-3θ.

shaded=5θ+3(π-θ)+θ=3π+3θ, unshaded=9π-(3π+3θ)=6π-3θ
4STEP 4

Set up the ratio and solve

From 3π+3θ = 8/13(6π-3θ): 39π+39θ = 48π-24θ, so 63θ = 9π and θ = π/7 — choice (B).

3π+3θ=8/13(6π-3θ) → 63θ=9π → θ=π/7=(B)
Answer
π/7
Plug θ=π/7 back in. Shaded =3π+3·π/7=(21π+3π)/7=24π/7. Unshaded =6π-3·π/7=(42π-3π)/7=39π/7. Their ratio is (24π/7)/(39π/7)=24/39=8/13, exactly as required. Also shaded plus unshaded is 24π/7+39π/7=63π/7=9π, the full disk, so nothing was lost or double-counted.
💡Key takeaway

When wedges meet at a center, a slice's area is just its angle's share of a full turn, so you can trade angles for areas and let one equation finish the job.

  • Split the picture into three rings
  • Name the angle and measure each wedge
  • Add up the shaded area
  • Set up the ratio and solve

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