AMC 10 · 2002 · #7

Grade 7 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
arc-measureratio-proportionarea-circles convert-to-algebra ↑ Prerequisites: arc-measureratio-proportion
📏 Medium solution 💡 2 insights
Problem
A 45° arc cut from circle A has the same length as a 30° arc cut from circle B. Using only that fact, find the ratio of circle A's area to circle B's area.

Pick an answer.

(A)
4/9
(B)
2/3
(C)
5/6
(D)
3/2
(E)
9/4
How to solve
Strategy Introduce a Variable

No lengths are given, so name what is missing: let r_A and r_B be the two radii (tool #4). Once each arc is written in terms of its radius, the 'equal length' fact becomes a single equation linking r_A and r_B. Tool #7 (Identify Subproblems) splits the work into two clean stages: first find the ratio of the radii, then turn that into the ratio of the areas — because area depends on the radius squared, these are genuinely different questions. Tool #3 (Eliminate Possibilities) guards the finish: the larger-angle arc belongs to the smaller circle, so circle A is smaller and the area ratio must be less than 1, which already throws out (D) 3/2 and (E) 9/4.

1STEP 1

Write each arc's length

An arc is its angle's share of 2πr: circle A gives 1/8 of its circumference, circle B 1/12.

arc_A=45/360 · 2π r_A=(π r_A)/4, arc_B=30/360 · 2π r_B=(π r_B)/6
2STEP 2

Set the arcs equal, find the radius ratio

Setting the arcs equal cancels π and gives 6r_A = 4r_B, so r_A/r_B = 2/3.

(π r_A)/4=(π r_B)/6 → 6r_A=4r_B → r_A/r_B=2/3
3STEP 3

Square the ratio to compare areas

Area scales as the radius squared, so the ratio is (2/3)² = 4/9, choice (A).

area_A/area_B=(π r_A²)/(π r_B²)=(r_A/r_B)²=(2/3)²=4/9 → (A)
Answer
4/9
Check the direction and the size. The 45° arc spans a bigger angle than the 30° arc, yet both lengths match, so circle A must be smaller to make its wider slice come out the same length — and indeed r_A/r_B=2/3 ＜ 1 and area ratio 4/9 ＜ 1, both saying A is smaller, which is consistent. The size is sensible too: squaring 2/3 shrinks it from 0.67 to about 0.44, and 4/9≈ 0.44 sits neatly between the smaller decoys, not near 1. The trap answer 2/3 is exactly the radius ratio left un-squared, and 9/4 flips the fraction the wrong way; the area ratio for the smaller circle over the larger has to be below 1.
💡Key takeaway

Equal arcs let you find the ratio of the radii; then square that ratio to get the ratio of the areas, because a circle's area grows with the radius squared.

  • Write each arc's length
  • Set the arcs equal, find the radius ratio
  • Square the ratio to compare areas

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