AMC 8 · 2007 · #22

Grade 6 geometry-2d

Archetype Pair-Sum Invariant └ Sub-archetype Rectilinear Perimeter Closure

coordinate-geometryspatial-visualizationpattern-recognition identify-subproblemspattern-recognition ↑ Prerequisites: coordinate-geometry

📏 Short solution 💡 2 insights

Problem

A lemming starts at a corner of a 10 × 10 square. It runs 6.2 meters along the diagonal toward the opposite corner, makes a 90^° right turn, and runs 2 more meters. From its final position, what is the average of the perpendicular distances to the four sides of the square?

Pick an answer.

(A)

(B)

4.5

(C)

(D)

6.2

(E)

AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Find an Invariant

The path detail (6.2 meters then a 90^° turn then 2 meters) is decoy. For any point inside a 10 × 10 square, the two distances to a pair of opposite sides always sum to 10 — that is the invariant. Tool #11 (Find an Invariant) catches this immediately. Tool #1 (Draw a Diagram) confirms the geometry: drop perpendiculars from the lemming's final spot to all four sides. Tool #4 (Introduce a Variable) lets us call the unknown coordinates x and y and watch them cancel.

1STEP 1

Put the start corner at the origin with sides on the axes, call the lemming's final spot (x, y), and drop perpendiculars to the four sides.

distances = x, 10 - x, y, 10 - y

2STEP 2

We don't need the actual values of x and y — only how each pairs with its complement, so write the four distances symbolically.

distance to left = x, distance to right = 10 - x, distance to bottom = y, distance to top = 10 - y

3STEP 3

Add the four distances: the x-terms cancel, the y-terms cancel, and the sum is always 20, wherever (x, y) lands.

x + (10 - x) + y + (10 - y) = 20

4STEP 4

The sum is 20, so the average of the four distances is $\frac{20}{4}$ = 5.

average =

\frac{20}{4}

= 5 → (C)

Answer

Pick any point inside the square and check. The center (5, 5) has distances 5, 5, 5, 5 averaging 5. The point (2, 9) has distances 2, 8, 9, 1 summing to 20 and averaging 5. The lemming's exact final position is irrelevant — every interior point gives the same average. The path-length numbers 6.2 and 2 are deliberate distractors, and the answer matches choice (C).

💡Key takeaway

Wherever the lemming lands inside a 10 × 10 square, the four perpendicular distances to the sides always add to 20, so the average is 5. The 6.2 and 2 in the path are distractors.

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