AMC 10 · 2008 · #13

Grade 8 geometry-2d

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

equilateral-trianglecoordinate-geometryarea-differencearea-triangles coordinate-geometryarea-difference ↑ Prerequisites: area-trianglescoordinate-geometry

📏 Long solution 💡 3 insights

Problem

An equilateral triangle sits on one side of a unit square with its apex inside. A vertical band of the square is marked off. Find the area of the band lying outside the triangle.

Pick an answer.

(A)

$\frac{12-5\sqrt3}{72}$

(B)

$\frac{12-5\sqrt3}{36}$

(C)

$\frac{\sqrt3}{18}$

(D)

$\frac{3-\sqrt3}{9}$

(E)

$\frac{\sqrt3}{12}$

How to solve

Strategy Draw a Diagram

Every condition in this problem is about position, so I put the square on coordinate axes with AD along the y-axis. Then 'distance from AD' becomes the x-coordinate and the vague band becomes the clean strip 1/3 ≤ x ≤ 2/3. The region R has a curved-sounding description but is really a strip with a bite taken out of it, so I compute the easy strip and subtract the bite. The bite itself is a pentagon, which splits into a rectangle plus a small triangle.

1STEP 1

Put the square on coordinates

Coordinates turn the band into a plain range.

dist((x,y), AD) = x for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1

2STEP 2

Locate E exactly

The apex sits at a height fixed by the equilateral condition.

h² + (1/2)² = 1² → h=√3/2, E=(1/2, 1-√3/2)

3STEP 3

Write the two slanted sides

The two slanted sides give the triangle's outline inside the band.

AE: y=1-√3 x, BE: y=1-√3 (1-x), △ ABE at x: 1-√3min(x,1-x) ≤ y ≤ 1

4STEP 4

The triangle never reaches the floor

The triangle never reaches the far side, so subtracting is clean.

[R] = [strip] - [△ ABE ∩ strip], [strip] = (2/3-1/3) · 1 = 1/3

5STEP 5

Cut the bite into rectangle plus triangle

The bite splits into a rectangle plus a triangle.

1/3·√3/3 + 1/2·1/3·√3/6 = √3/9+√3/36 = 5√3/36

6STEP 6

Subtract and match a choice

Subtracting gives (12-5√3)/36, choice (D).

[R] = 1/3-5√3/36 = (12-5√3)/36 ≈ 0.0928

Answer

(12-5√3)/36

Two checks. Size: (12-5√3)/36≈ 0.0928 is positive and clearly smaller than the whole strip 1/3≈ 0.333, exactly as a leftover should be. Proportion: the bite I removed is 5√3/36≈ 0.2406 out of the triangle's total area √3/4≈ 0.4330, which is about 56% of the triangle. That is comfortably more than 1/3, which is right, because the middle strip catches the triangle exactly where it is widest. If the computation had returned less than a third of the triangle, something would be wrong.

💡Key takeaway

Put coordinates on the picture so that distance from a side becomes an x-value, then take the easy strip and subtract the triangle's bite instead of chasing the leftover shape directly.

Put the square on coordinates
Locate E exactly
Write the two slanted sides
The triangle never reaches the floor
Cut the bite into rectangle plus triangle
Subtract and match a choice