Problem 18 · 2022 AMC 8
Hard
Geometry & Measurement
areaarea-decomposition
The midpoints of the four sides of a rectangle are (−3, 0), (2, 0), (5, 4), and (0, 4). What is the area of the rectangle?
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Answer: C — 40.
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Hint 1 of 2
You're handed the midpoints, not the corners — so reach for the fact about midpoints. Joining the midpoints of any quadrilateral's sides makes a parallelogram, and it always has exactly half the area of the original.
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Hint 2 of 2
Plot the four midpoints and find that inner parallelogram's area (base × height), then double it to recover the rectangle.
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Approach: Varignon — the midpoint parallelogram has half the parent's area
- Insight: don't hunt for the rectangle's corners. Connecting the midpoints of any quadrilateral gives a parallelogram (Varignon's theorem) whose area is always half the original — intuitively, each corner triangle you slice off to go from quadrilateral to inner parallelogram removes a quarter of the area, and the four removed quarters total half.
- Plot the midpoints: the base from (−3,0) to (2,0) has length 5, and the height (y from 0 to 4) is 4, so the inner parallelogram has area 5 × 4 = 20.
- Double it: rectangle area = 2 × 20 = 40.
Another way — the segments joining opposite midpoints ARE the rectangle's sides:
- A segment connecting the midpoints of two opposite sides of a rectangle runs straight across and has the same length as the pair of sides it's parallel to — so the two such segments give the rectangle's width and height directly.
- Opposite midpoints: (−3,0)–(5,4) gives a segment of length √(82 + 42) = √80; (2,0)–(0,4) gives √(22 + 42) = √20.
- These are perpendicular (a rectangle's adjacent sides), so area = √80 × √20 = √1600 = 40.
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