Problem 24 · 2024 AMC 8
Hard
Geometry & Measurement
areaarea-decomposition
Show answer
Answer: B — h = 5.
Show hints
Hint 1 of 2
The dip between the peaks is exactly where the two mountains OVERLAP. So if you treat each mountain as a full triangle and add them, you've counted that dip twice — the setup for inclusion–exclusion.
Still stuck? Show hint 2 →
Hint 2 of 2
Handy fact: a 45-45-90 mountain triangle of peak height H has base 2H, so its area is 12(2H)(H) = H2. The dip is a third such triangle of height h.
Show solution
Approach: inclusion–exclusion on three 45-45-90 triangles
- First, a clean area shortcut: each mountain is a right-isoceles triangle (90° peak, 45° base angles), so its base is 2×(its height) and its area is 12(2H)(H) = H2. Heights 8 and 12 give areas 64 and 144.
- If you just add 64 + 144, you double-count the V-dip where the two mountains overlap — and that dip is itself a 45-45-90 triangle of height h, area h2. So the true artwork area is 64 + 144 − h2.
- Set that to the given 183: 208 − h2 = 183 → h2 = 25 → h = 5. This transfers: when two regions overlap, area(A) + area(B) − area(overlap) = total — inclusion–exclusion turns a tricky shape into three simple ones.
Mark:
· log in to save