AMC 10 · 2014 · #23

Grade 8 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Overlapping Translates

paper-foldingarea-trianglesarea-difference spatial-visualizationreflection-unfolding ↑ Prerequisites: area-triangles

📏 Medium solution 💡 2 insights 📊 Diagram

Problem

A rectangle is √3 times as long as it is wide and has area A. Its two long edges are each split into three equal parts, and a straight crease is drawn from the first split point on one long edge to the second split point on the opposite long edge. The sheet is folded flat along this crease, and the folded silhouette has area B. Find the ratio B/A.

Pick an answer.

(A)

$\frac{1}{2}$

(B)

$\frac{3}{5}$

(C)

$\frac{2}{3}$

(D)

$\frac{3}{4}$

(E)

$\frac{4}{5}$

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Visualize Spatial Relationships

The crease and the fold are a physical, spatial action, so Tool #17 (Visualize Spatial Relationships) is the anchor: you must picture the flap flipping over the crease before any number helps. Tool #1 (Draw a Diagram) turns that mental picture into coordinates, so 'where does the corner land' becomes something you can compute. Tool #7 (Identify Subproblems) splits the task into two clean pieces: first see that B=A-overlap, then measure the overlap. Tool #4 (Introduce a Variable) fixes a convenient width of 1 so every area is a concrete number.

1STEP 1

Fix a width and set coordinates

Take the width as 1 with the sheet on axes, so the length is √3 and A=√3; the crease joins P=(√3/3,0) to Q=(2√3/3,1).

A=√3, P=(√3/3,0), Q=(2√3/3,1)

2STEP 2

Turn folding into area minus overlap

A fold is a mirror, so B=A-overlap. Crease PQ perpendicularly bisects the diagonal D=(0,1) to R=(√3,0), so D lands on R.

B=A-overlap; slope_PQ…lope_DR=√3·(-1/√3)=-1

3STEP 3

Measure the doubly-covered triangle

The double layer is triangle PQR with base PR=2√3/3 on the bottom edge and height 1, so its area is √3/3.

[PQR]=1/2·2√3/3 · 1=√3/3

4STEP 4

Take the ratio B over A

Subtract: B=√3-√3/3=2√3/3. Dividing by A=√3 cancels the radical and leaves 2/3, choice (C).

B/A=(√3-√3/3)/√3=2√3/3/√3=2/3 (C)

Answer

2/3

The fold only tucks one corner inward, so it should erase a modest slice of paper, not most of it; a ratio comfortably above 1/2 and below 1 is expected, and 2/3 fits. Cross-check the overlap triangle by its sides: the two slanted sides PQ and QR each measure √((√3/3)²+1²)=2/√3=2√3/3, and the base PR is also 2√3/3, so triangle PQR is equilateral. That is exactly the symmetric shape a 60° crease should fold up, which supports the overlap area √3/3 and the final 2/3.

💡Key takeaway

Folding a corner over just hides one overlapping triangle, and here that lost triangle is one-third of the sheet, so two-thirds of the paper is left: B/A=2/3.

Fix a width and set coordinates
Turn folding into area minus overlap
Measure the doubly-covered triangle
Take the ratio B over A

Fix a width and set coordinates

Turn folding into area minus overlap

Measure the doubly-covered triangle

Take the ratio B over A

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