Problem 18 · 2000 AMC 8
Hard
Geometry & Measurement
area-decompositionperimeter
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Answer: E — Same area, but quadrilateral I has the smaller perimeter.
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Hint 1 of 2
The answer choices split into 'compare area' and 'compare perimeter,' so settle area FIRST — and don't eyeball it. On a geoboard you can find area exactly (slice into triangles, or use base × height).
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Hint 2 of 2
Equal area does NOT force equal perimeter — a shape can enclose the same space with more boundary. Once areas tie, compare the sides one by one: the shapes share a pair of equal slants, so look only at the *other* sides.
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Approach: lock down area exactly, then compare the differing sides
- Area first. Region I is a parallelogram of base 1 and height 1, so area = 1. Region II splits into two triangles, each base 1 and height 1, area ½ + ½ = 1. The areas are *equal* — so it's down to perimeter.
- Both shapes have a matching pair of slant sides (each spanning 1 across and 1 up, length √2). Their remaining sides differ: region I's other two sides are plain unit segments (length 1 each), while region II's remaining side is a long diagonal clearly stretching more than 1.
- So region II carries more boundary — its perimeter is larger, which means region I's perimeter is the *smaller* one: choice E.
- The big idea: area and perimeter are independent — same area can hide very different perimeters (a key reason 'compare the figures' problems list both). Pin down the one that's easy to compute exactly, then reason about the other.
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