Problem 9 · 1995 AJHSME
Hard
Geometry & Measurement
tangent-circles
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Answer: C — 32.
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Hint 1 of 2
Don't reach for circle formulas — area of a rectangle only needs its width and height, and the circles are just there to MEASURE those lengths. The radius is your ruler.
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Hint 2 of 2
'Circle Q passes through P and R' is the secret length clue: P and R sit on circle Q, so PQ and QR each equal Q's radius. Tangency to the top and bottom edges gives the height.
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Approach: use the radius as a ruler to measure the rectangle's sides
- Diameter 4 means radius 2. The circles are tangent to the top and bottom edges, so the rectangle's height is exactly one diameter: 4.
- For the width, use the clue 'Q passes through P and R': that puts P and R on circle Q, so PQ = QR = 2 (a radius), giving PR = 4. The left and right circles are tangent to the side edges, adding one radius (2) beyond P and beyond R.
- Width = 2 + PR + 2 = 2 + 4 + 2 = 8. Area = 8 × 4 = 32.
- The reusable idea: when a figure is built from tangent circles, every key length is a whole number of radii. Count radii along each edge instead of computing anything.
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