Competition · AMC preparation · step 4 of 4

AMC 8 · 2024 · #24

Grade 6 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Overlapping Translates
area-trianglesperfect-squares area-differencecasework ↑ Prerequisites: area-trianglesmulti-digit-arithmetic
📏 Long solution 💡 5 insights 📊 Diagram
Problem
A stained-glass artwork is shaped like two overlapping mountain triangles. Each peak is a 90° angle, and each mountain's two straight sides meet the ground at 45°. The left peak is 8 feet high, the right peak is 12 feet high, and the total artwork covers 183 square feet. The two mountains overlap, and their inner sides cross at a point h feet above the ground. Find h.

Pick an answer.

(A)
4
(B)
5
(C)
$4\sqrt{2}$
(D)
6
(E)
$5\sqrt{2}$

AMC 8 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The artwork is the union of two overlapping mountain-triangles, so Tool #1 (Draw a Diagram) is the starting point — we extend each mountain's inner side back down to the ground so we can see two complete large triangles overlapping in a smaller triangle. Tool #7 (Identify Subproblems) breaks the problem into three friendly area pieces (big triangle, other big triangle, the overlap). To find the bases of the mountains from their heights, Tool #9 (Easier Related Problem) lets us replace each mountain with a tiny 45° unit triangle and notice that a 45° slope means "one over, one up," so the base is exactly twice the peak height. After we get h² = 25, Tool #6 (Guess and Check) finds h = 5 instantly, and Tool #3 (Eliminate Possibilities) confirms that 5 is choice (B) while the √(2) choices and the others are not consistent with the area equation.

1STEP 1

Extend the slopes to the ground

Extend each inner side down to the ground: two full triangles overlap in a small one, so area_left + area_right - overlap = 183.

Area_left + Area_right - Area_overlap = 183
2STEP 2

Find each base from its height

Each mountain is an isosceles right triangle, so its base is twice the peak height: base_left = 16, base_right = 24.

base_left = 2 × 8 = 16, base_right = 2 × 12 = 24
3STEP 3

Find the two triangle areas

Use base × height ÷ 2 for each big triangle: area_left = 64 and area_right = 144 square feet.

Area_left = 1/2 · 16 · 8 = 64, Area_right = 1/2 · 24 · 12 = 144
4STEP 4

Write the overlap in terms of h

The overlap is an upside-down isosceles right triangle of height h and base 2h, so its area is h².

Area_overlap = 1/2 · (2h) · h = h²
5STEP 5

Set up inclusion-exclusion

Substitute into 64 + 144 - h² = 183 and simplify: h² = 25.

64 + 144 - h² = 183 → h² = 208 - 183 = 25
6STEP 6

Solve for h

Since h is a positive height, h = 5 feet, which is choice (B); the √(2) options give h² = 32 or 50, so they are out.

5² = 25 → h = 5 → (B)
Answer
5
Sanity check the size. The two big triangles together have area 64 + 144 = 208 square feet, and the artwork is 183, so the overlap takes away 25 square feet — a small bite, exactly what we'd expect from a slim wedge near the middle. The overlap peak sits at h = 5 feet, which is below both mountain peaks (8 and 12) and above the ground — geometrically possible. Plugging back: 64 + 144 - 5² = 208 - 25 = 183. The units are all square feet, consistent throughout. The √(2) choices were red herrings tied to slanted-side lengths, not the vertical height the problem actually asked for.
💡Key takeaway

This AMC 8 problem only needs Grade 6 triangle-area thinking you already know!

  • Extend the slopes to the ground
  • Find each base from its height
  • Find the two triangle areas
  • Write the overlap in terms of h
  • Set up inclusion-exclusion
  • Solve for h

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