Problem 25 · 2000 AMC 8
Stretch
Geometry & Measurement
area-decomposition
The area of rectangle ABCD is 72. If point A and the midpoints of sides BC and CD are joined to form a triangle, the area of that triangle is
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Answer: B — 27.
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Hint 1 of 2
The slanted triangle is hard to measure head-on, but the three corners it leaves behind are clean right triangles. Cut those away from the whole rectangle instead — and you never need the actual dimensions.
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Hint 2 of 2
Each corner triangle is ½ · (its two legs), and each leg is either a full side or a half-side of the rectangle — so each corner is a simple fraction (¼ or ⅛) of the rectangle. Add the fractions, subtract from 1.
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Approach: subtract the corner triangles as fractions of the whole
- The target triangle is what's left after slicing off the three right triangles in the corners. Work in fractions of the whole rectangle so the unknown dimensions cancel:
- Corner B: legs = full top + half of the right side ⇒ ½ · 1 · ½ = ¼ of the rectangle. Corner D: full left side + half the bottom ⇒ another ¼. Corner C: half-right + half-bottom ⇒ ½ · ½ · ½ = ⅛.
- Corners take ¼ + ¼ + ⅛ = ⅝, so the triangle is the remaining ⅜. Area = ⅜ · 72 = 27.
- You'll see it again: for a triangle drawn inside a rectangle using corners and midpoints, don't hunt for a base and height — subtract the corner right triangles as fractions of the whole. The area ratio is fixed no matter the rectangle's actual shape.
Another way — drop in coordinates:
- Pick easy dimensions with area 72, say width 12 and height 6: A(0,6), B(12,6), C(12,0), D(0,0). Midpoint of BC is M(12,3); midpoint of CD is N(6,0).
- Shoelace on A(0,6), M(12,3), N(6,0): area = ½|0(3−0) + 12(0−6) + 6(6−3)| = ½|0 − 72 + 18| = ½·54 = 27.
- Because the answer is a fixed fraction of the area, any width×height = 72 gives the same 27 — handy as a check.
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