Problem 7 · 2001 AMC 8
Medium
Geometry & Measurement
area
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.
What is the number of square inches in the area of the small kite?
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Answer: A — 21 square inches.
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Hint 1 of 2
The kite's diagonals cross at a right angle (one across, one up-and-down on the grid) — that perpendicular crossing is what unlocks an easy area.
Still stuck? Show hint 2 →
Hint 2 of 2
When a quadrilateral has perpendicular diagonals, its area is ½ × (one diagonal) × (other diagonal). Read both lengths off the grid.
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Approach: area of a kite = half the product of the diagonals
- Count along the grid: the horizontal diagonal is 6 inches and the vertical diagonal is 7 inches, and they meet at a right angle.
- For perpendicular diagonals, Area = ½ × d₁ × d₂ = ½ × 6 × 7 = 21 square inches.
- Why it works: the two diagonals slice the kite into four right triangles, and gathering their areas always gives ½·d₁·d₂. The same formula handles any rhombus or square (just diagonals) — keep it in your toolkit.
Another way — two triangles sharing the horizontal diagonal:
- The horizontal diagonal (length 6) splits the kite into a top triangle and a bottom triangle.
- Their heights are the two pieces of the vertical diagonal, totaling 7, so area = ½ × 6 × 7 = 21 square inches.
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