Problem 9 · 2001 AMC 8
Medium
Geometry & Measurement
area-fractionspatial-reasoning
To promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. The kites look like the one in the diagram. For her small kite Genevieve draws the kite on a one-inch grid (shown below). For the large kite she triples both the height and width of the entire grid.
The large kite is covered with gold foil. The foil is cut from a rectangular piece that just covers the entire grid. How many square inches of waste material are cut off from the four corners?
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Answer: D — 189 square inches.
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Hint 1 of 2
The kite's diagonals are exactly the rectangle's width and height — so kite area is ½ × width × height, which is half the rectangle. No need to measure the four corners separately.
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Hint 2 of 2
Waste = rectangle − kite, but since the kite IS half the rectangle, the waste is just the other half.
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Approach: waste = rectangle − kite, and the kite is half the rectangle
- The large rectangle covers the whole grid: 18 × 21 = 378 square inches.
- Here's the shortcut: the kite's diagonals span the full 18 and 21, so its area is ½ × 18 × 21 — exactly half the rectangle. The cut-off corners are therefore the other half: 378 ÷ 2 = 189 square inches.
- Spotting that a shape is a clean fraction of its bounding box saves the messier work of computing four corner triangles one by one — look for that whenever a figure sits snugly inside a rectangle.
Another way — subtract the kite explicitly:
- Kite area = ½ × 18 × 21 = 189 square inches.
- Waste = rectangle − kite = 378 − 189 = 189 square inches — confirming the two halves match.
Mark:
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