Competition · AMC preparation · step 4 of 4
AMC 8 · 2012 · #5
Grade 4 geometry-2d
Pick an answer.
AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Walk around the boundary
Walk the boundary once and mark each vertical edge with an up-arrow or a down-arrow; ignore the horizontal edges for now.
Recognizing that a closed rectilinear polygon is made only of horizontal and vertical edges is a Grade 3 shape-attribute idea.
3.G.A.1Draw A DiagramList the upward edges
Tracing the boundary, the four edges that go up carry the labels 6, 1, 2, 1.
Sorting edges into "up" and "down" piles is the systematic-list move; we are organizing the perimeter data.
3.MD.D.8Make A Systematic ListAdd the upward lengths
Adding the up-edge lengths gives a total upward distance of 10 cm.
Adding labeled whole-number lengths is exactly the Grade 4 multi-step word-problem skill.
4.OA.A.3Make A Systematic ListList the downward edges
Tracing onward, the downward edges carry the labels 1, 1, 1, 2, X (X is the unknown drop back to the start).
Same systematic-list move, applied to the opposite direction so nothing is double-counted.
3.MD.D.8Make A Systematic ListAdd the downward lengths
The known downs sum to 5, so the full downward total is 5 + X.
Combining labeled lengths into a single expression sets up the closure equation.
4.OA.A.3Make A Systematic ListMatch the up and down totals
Closure forces up = down, so 10 = 5 + X and X = 5.
Finding a missing side length from the rest of the boundary is exactly the Grade 3 perimeter standard, just applied direction-by-direction.
In this closed right-angled figure, the upward vertical edges add up to the same total length as the downward vertical edges.
▸ Why?
Tracing the whole boundary is a single closed loop, so you finish at the exact point you started, ending at the same height with no overall rise or fall.
▸ Why?
Only the vertical edges change your height, and an upward edge lifts you by its length while a downward edge lowers you by that same kind of length, so an up move and a down move are opposite actions that cancel each other.
▸ Why?
Since the ups and downs cancel to leave you back at the start, the whole distance you rose has to match the whole distance you fell, or else you would end up above or below where you began.
▸ Why?
The whole rise is just all the upward edge lengths joined together and the whole fall is all the downward edge lengths joined together, each set of edges combining with no gaps or overlaps into its own total.
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.
- Walk around the boundary
- List the upward edges
- Add the upward lengths
- List the downward edges
- Add the downward lengths
- Match the up and down totals
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