Problem 10 · 2025 AMC 8
Medium
Geometry & Measurement
areaarea-decompositiontransformations
Show answer
Answer: D — 23.75 square inches.
Show hints
Hint 1 of 2
Don't try to find the funny outline directly. If you just add both rectangles' areas, what region gets counted twice?
Still stuck? Show hint 2 →
Hint 2 of 2
Area covered = (one rectangle) + (other rectangle) − (the overlap). So you only need the overlap's area. It's a square — and the pivot at the midpoint of DC fixes its side at 2.5.
Show solution
Approach: inclusion–exclusion (add both, subtract the double-count)
- The shape covered is awkward, but its area isn't: add both rectangles and subtract the part you counted twice. Each rectangle is 5 × 3 = 15.
- The overlap: the rotation pivots at the midpoint of DC, so along DC the shared strip is half of 5 = 2.5, and a quarter-turn makes the overlap a 2.5 × 2.5 square — area 2.52 = 6.25.
- Covered area = 15 + 15 − 6.25 = 23.75.
- Why this transfers: whenever two regions overlap, |A or B| = |A| + |B| − |A and B| — the overlap must be subtracted once because adding counts it twice. This inclusion–exclusion idea saves you from ever computing a messy combined outline.
Mark:
· log in to save