AMC 8 · 2008 · #18

Grade 7 geometry-2d

Archetype Path Length Comparison & Shortest Path └ Sub-archetype Sum Known Path Segments

perimeterarea-circlesfraction-arithmeticspatial-visualization path-length-comparisonidentify-subproblems ↑ Prerequisites: area-circlesfraction-arithmetic

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Two circles share the same center. The small one has radius 10 meters and the large one has radius 20 meters. An aardvark walks from A at the top of the large circle to K at the right of the large circle, following the path of arrows in the figure. Find the total length of that path.

Pick an answer.

(A)

$10\pi+20$

(B)

$10\pi+30$

(C)

$10\pi+40$

(D)

$20\pi+20$

(E)

$20\pi+40$

AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Break the Problem into Subproblems

The path is one curvy trip, but it is built from six clean pieces: three quarter-arcs and three straight segments. Tool #7 (Break the Problem into Subproblems) says to find the length of each piece by itself, then add. Tool #1 (Draw a Picture) helps us label each piece with its endpoints in coordinates so we know which circle the arc sits on and which radius the segment uses.

1STEP 1

Follow the arrows from A to K and split the trip into six pieces: three quarter-arcs and three straight segments.

path = arc₁ + seg₁ + arc₂ + seg₂ + arc₃ + seg₃

2STEP 2

The first piece is a quarter of the large circle (circumference 40π), giving an arc of 10π.

arc₁ =

\frac{1}{4}

(2π · 20) =

\frac{1}{4}

(40π) = 10π

3STEP 3

Each small arc is a quarter of the small circle (circumference 20π), so each one is 5π.

arc₂ = arc₃ =

\frac{1}{4}

(2π · 10) = 5π

4STEP 4

Each x-axis segment is 20-10=10; the y-axis segment is the small circle's diameter, 20.

seg₁ = 10, seg₂ = 20, seg₃ = 10

5STEP 5

Group π-terms and constants: 10π+5π+5π=20π and 10+20+10=40, so the total is 20π+40 → (E).

10π + 10 + 5π + 20 + 5π + 10 = (10π + 5π + 5π) + (10 + 20 + 10) = 20π + 40 → (E)

Answer

20π+40

Sanity check the size. The path includes one big quarter-arc on the outer circle (10π ≈ 31.4) plus two small quarter-arcs (5π + 5π ≈ 31.4) plus three straight pieces (10 + 20 + 10 = 40). Total ≈ 62.8 + 40 ≈ 102.8 m, which matches 20π + 40 ≈ 102.8. The full perimeter of the large circle is only 40π ≈ 125.7 m, so a longer-than-half-circle answer with extra straight pieces is reasonable. Among the choices, only (E) has the 20π coefficient that comes from one big quarter-arc plus two small quarter-arcs combined.

💡Key takeaway

A curvy path becomes easy when you cut it into pieces — each arc is a quarter of its circle and each straight piece is a difference of coordinates.

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