Problem 4 · 1985 AJHSME
Medium
Geometry & Measurement
bounding-rect-minus-notch
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Answer: C — 46.
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Hint 1 of 2
The shape has one corner pushed IN (the bite near E and F). Instead of chopping it into pieces, imagine filling that bite to make a full rectangle — then take the bite back out. What is the full rectangle?
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Hint 2 of 2
This is the bounding-box trick: enclose any rectilinear shape in the smallest rectangle that contains it, then subtract the rectangular pieces that aren't part of the shape. One subtraction beats slicing into several rectangles.
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Approach: bounding rectangle minus the notch
- Fill the bite to complete the full rectangle: it is 6 wide (the top AB) by 9 tall (the right side BC), so its area is 6 × 9 = 54.
- The bite that was removed is itself a small rectangle. Its width is 6 − 4 = 2 (top width minus bottom width DC) and its height is 9 − 5 = 4 (right side minus left side AF), so its area is 2 × 4 = 8.
- Polygon = full rectangle − bite = 54 − 8 = 46.
- Why this transfers: any L-shape, T-shape, or staircase is just a rectangle with rectangular bites taken out — find the missing side lengths by subtracting the parts you know, then subtract areas.
Another way — split into two rectangles:
- Cut horizontally at the level of F/E. Top piece: 6 wide × 5 tall = 30. Bottom piece: 4 wide × (9 − 5) = 4 × 4 = 16.
- 30 + 16 = 46 — same answer, built up instead of cut down.
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