Problem 13 · 2005 AMC 8
Medium
Geometry & Measurement
rectilinear-completion
The area of polygon ABCDEF is 52 with AB = 8, BC = 9 and FA = 5. What is DE + EF?
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Answer: C — 9.
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Hint 1 of 2
This L-shape is a full rectangle with a smaller rectangle bitten out of the corner. Fill the bite back in — the big rectangle is easy, and the bite carries the unknowns DE and EF.
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Hint 2 of 2
The missing bite's area = (big rectangle) − (given area). One side of the bite is forced by the two side lengths you already know — that hands you the other side.
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Approach: complete the L to a rectangle, subtract the bite
- Extend the short edges to close off the corner. The full bounding rectangle is 8 × 9 = 72. The L has area 52, so the bitten-out rectangle has area 72 − 52 = 20.
- The bite's vertical side is the gap between the two sides BC = 9 and FA = 5, namely ED = 9 − 5 = 4.
- So its horizontal side is EF = 20 ÷ 4 = 5, and DE + EF = 4 + 5 = 9.
- Why this transfers: any rectilinear (right-angle-only) polygon is a rectangle plus or minus smaller rectangles — 'complete to the bounding box and subtract' turns a scary shape into easy arithmetic.
Another way — edges close around, not just area:
- In a closed rectilinear path, the horizontal edges going right must equal those going left: AB = FE + DC, and the vertical edges balance the same way.
- Combined with area 52, the same numbers fall out: ED = 4, EF = 5, giving 9. Useful when you're handed perimeters instead of an area.
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