AMC 10 · 2014 · #19

Grade 8 geometry-3d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length

coordinate-geometrysimilar-trianglespythagorean-theorem spatial-visualizationidentify-subproblems ↑ Prerequisites: coordinate-geometrysimilar-triangles

📏 Short solution 💡 2 insights 📊 Diagram

Problem

Four cubes with edge lengths 1, 2, 3, and 4 are stacked in a staircase tower that shares one vertical corner post. X is the top corner of the smallest cube (top of the post) and Y is the far bottom corner of the largest cube. Find how much of the straight segment XY lies inside the cube whose edge is 3.

Pick an answer.

(A)

$\dfrac{3\sqrt{33}}5$

(B)

$2\sqrt3$

(C)

$\dfrac{2\sqrt{33}}3$

(D)

(E)

$3\sqrt2$

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

How to solve

Strategy Visualize Spatial Relationships

The picture is a 3D staircase, so Tool #17 (Visualize Spatial Relationships) is the anchor: we must picture where X, Y, and the middle cube sit in space before any numbers help. Tool #1 (Draw a Diagram) turns that mental picture into xyz-coordinates so lengths become computable. Tool #7 (Identify Subproblems) splits the task into two clean pieces — first the full length of XY, then the slice inside the edge-3 cube. Tool #4 (Introduce a Variable) lets us slide a point along XY to see exactly which part of it is at the height of the edge-3 cube.

1STEP 1

Set coordinates and place X and Y

Put the shared post on the z-axis: the cubes fill heights 0-4, 4-7, 7-9, 9-10, so X=(0,0,10) and Y=(4,4,0).

X=(0,0,10), Y=(4,4,0)

2STEP 2

Find the full length of XY

X to Y moves 4 across, 4 along, and 10 down, so 3-D Pythagoras gives XY=√(4²+4²+10²)=√(132)=2√(33).

XY=√(4²+4²+10²)=√(132)=2√(33)

3STEP 3

Take the slice at the edge-3 cube's height

The edge-3 cube covers 3 of the 10-unit drop and x,y stay under 3, so its slice is 3/10·2√(33)=3√(33)/5, choice (A).

3/10 · 2√(33)=3√(33)/5 (A)

Answer

3√(33)/5

The full segment is 2√(33)≈ 11.5, and the edge-3 cube is 3 of the tower's 10 units tall, so its slice should be a bit under a third of 11.5, roughly 3.4. The answer 3√(33)/5≈ 3.45 matches. It is also correctly larger than the edge-1 or edge-2 slices would be and smaller than the edge-4 slice, since a taller cube captures a longer piece of the same slanted line. The footprint check (x,y reaching only 2.4 ＜ 3) confirms the segment never leaks out a side face, so the whole height band counts.

💡Key takeaway

A straight slanted line loses length in step with height, so the cube that is 3 tall out of 10 swallows exactly 3/10 of the segment: 3/10 · 2√(33)=3√(33)/5.

Set coordinates and place X and Y
Find the full length of XY
Take the slice at the edge-3 cube's height

Set coordinates and place X and Y

Find the full length of XY

Take the slice at the edge-3 cube's height

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