AMC 10 · 2021 · #15

Grade 8 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation

equilateral-trianglearea-trianglesthirty-sixty-ninety-trianglearea-difference identify-subproblemsarea-difference ↑ Prerequisites: area-trianglesequilateral-triangle

📏 Long solution 💡 4 insights 📊 Diagram

Problem

A pentagon is drawn together with two interior points, using eleven line segments, and every segment has length 2. The pentagon's area can be written as one square root plus another. Find the sum of the two numbers under those roots.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Draw a Diagram

Tool #1 (Draw a Diagram) — sketch the pentagon and mark all 11 length-2 segments. The drawing makes the four equilateral sub-triangles (△ ABF, △ BCF, △ AGE, △ GDE) jump out, and from there the 120° angles at B and E are visible (two adjacent 60° angles). Tool #7 (Identify Subproblems) — split the pentagon into the three triangles △ ABC, △ AED, and △ ACD along the diagonals AC and AD. Each sub-area is computable on its own with elementary triangle facts, and the three add to the answer. Tool #17 (Visualize) supports the symmetry observation that lets us only compute △ ABC once (since △ AED is its mirror image).

1STEP 1

Spot the equilateral triangles

Equal lengths make equilateral triangles.

AB = BF = AF = 2 → △ ABF equilateral, ∠ ABF = 60°; similarly for the other three.

2STEP 2

Fill in the angles

Two equilateral triangles meet at 120 degrees.

∠ ABC = 60° + 60° = 120°; ∠ AED = 120°

3STEP 3

Cut into three triangles

Split the pentagon into three triangles.

[ABCDE] = [ABC] + [ACD] + [AED]

4STEP 4

Measure the first triangle

Two sides and the angle give the area.

[ABC] = 1/2(2)(2)sin 120° = 2 · √(3)/2 = √(3)

5STEP 5

Measure its mirror

Its mirror matches.

[AED] = √(3)

6STEP 6

Find the diagonal

The law of cosines gives the diagonal.

AC² = 4 + 4 + 4 = 12, AC = AD = 2√(3)

7STEP 7

Measure the middle triangle

Being isosceles makes the height easy.

AM = √(12 - 1) = √(11); [ACD] = 1/2(2)(√(11)) = √(11)

8STEP 8

Add them up

Adding the numbers under the roots gives 23.

[ABCDE] = √(12) + √(11) → m + n = 12 + 11 = 23 → (D)

Answer

Sanity-check magnitudes. √(12) ≈ 3.46 and √(11) ≈ 3.32, so the pentagon area is about 6.78. The pentagon has "width" roughly |BE| ≈ 3 (centers of two equilateral triangles separated horizontally) and "height" roughly 2 — so an area near 7 is in the right ballpark. Also each ear triangle (area √(3) ≈ 1.73) is plausible for a triangle with two sides of 2 and a 120° apex. The form √(m) + √(n) with m, n both small integers near 12 matches the answer-choice spread (20 to 24), and 23 is precisely (D).

💡Key takeaway

This AMC 12 problem only needs Grade 8 Pythagorean reasoning you already know! Cut the pentagon with diagonals AC and AD — the two outer triangles have 120° apexes and area √(3) each, and the middle isoceles triangle has altitude √(11) and area √(11). Total = 2√(3) + √(11) = √(12) + √(11), so m + n = 23.