AMC 8 · 2012 · #5

Grade 4 geometry-2d

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

perimeterspatial-visualizationlinear-equations-one-var convert-to-algebraidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticperimeter

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A closed shape has only right angles, so every edge is either horizontal or vertical. All the side lengths are labeled in centimeters except one vertical edge marked X. The figure is not to scale, so we must trust the labels, not the picture. Find X.

Pick an answer.

(A)

$hspace{.05in}1$

(B)

$hspace{.05in}2$

(C)

$hspace{.05in}3$

(D)

$hspace{.05in}4$

(E)

$hspace{.05in}5$

AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

Because the figure closes up, if we walk all the way around the perimeter we must end up exactly where we started. That means the total upward distance must equal the total downward distance (otherwise we would float above or sink below the start). Tool #1 (Draw a Diagram) says: copy the figure and mark each vertical edge with an up-arrow or a down-arrow as you trace the boundary. Tool #2 (Make a Systematic List) says: write the up-lengths in one column and the down-lengths in another, then add each column. Setting up-total = down-total gives a single equation for X — no algebra harder than basic subtraction.

1STEP 1

Walk the boundary once and mark each vertical edge with an up-arrow or a down-arrow; ignore the horizontal edges for now.

2STEP 2

Tracing the boundary, the four edges that go up carry the labels 6, 1, 2, 1.

up-edges: 6, 1, 2, 1

3STEP 3

Adding the up-edge lengths gives a total upward distance of 10 cm.

6 + 1 + 2 + 1 = 10 cm

4STEP 4

Tracing onward, the downward edges carry the labels 1, 1, 1, 2, X (X is the unknown drop back to the start).

down-edges: 1, 1, 1, 2, X

5STEP 5

The known downs sum to 5, so the full downward total is 5 + X.

1 + 1 + 1 + 2 + X = 5 + X cm

6STEP 6

Closure forces up = down, so 10 = 5 + X and X = 5.

10 = 5 + X → X = 10 - 5 = 5 cm → (E)

Answer

hspace{.05in}5

Quick horizontal sanity check on the same closed-loop idea: right-going edges are 3, 1, 2, 2, 2 = 10 and left-going edges are 3, 2, 1, 4 = 10. They match, so the figure really does close, which means our vertical equation is being read the right way. The answer X = 5 is one of the offered choices and the largest, which fits the picture where X is one of the longer vertical drops.

💡Key takeaway

This AMC 8 problem only needs Grade 4 thinking: if a shape closes up, the total "up" must equal the total "down" — so the missing side is whatever balances the two columns.

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