Competition · AMC preparation · step 4 of 4

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)
2
(B)
4.5
(C)
5
(D)
6.2
(E)
7

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

Set up coordinates

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

Name the four distances

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

Spot the constant sum

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

Divide by 4 for the average

The sum is 20, so the average of the four distances is 204\frac{20}{4} = 5.

average = 20/4 = 5 → (C)
Answer
5
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.

  • Set up coordinates
  • Name the four distances
  • Spot the constant sum
  • Divide by 4 for the average

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