AMC 10 · 2012 · #2
Grade 3 geometry-2dPick an answer.
AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The whole question is about a shape being cut, so Tool #1 (Draw a Diagram) makes the situation visible: sketch the 8 by 8 square and the single cut. Tool #17 (Visualize Spatial Relationships) keeps track of which side gets shortened and which stays the same. Tool #3 (Eliminate Possibilities) guards against the trap choice (D) 4 by 4, which comes from wrongly halving both sides.
Draw the square and its cut
Sketch the square with every side 8, then one straight cut through the middle, parallel to one pair of sides.
A single straight cut can only shorten the shape in one direction, not both.
3.G.A.2Draw A DiagramHalve one side, keep the other
The cut halves the pair of sides it crosses: 8 ÷ 2 = 4. The sides along the cut are untouched, so they stay 8.
Cutting in half divides one length by 2 while the crosswise length rides along unchanged.
Cutting in half divides one length by two while the crosswise length rides along unchanged.
▸ Why?
Half is one of two equal shares, so the cut length becomes exactly one of those shares.
▸ Why?
The two pieces together rebuild the square, so nothing can be lost in the other direction.
Read off and match the dimensions
So each piece is 4 by 8, choice (E). Traps: 4 by 4 halves both sides, 2 by 8 splits one side into four.
Two equal shares of the square must together rebuild it, and 4 × 8 twice gives back the 8 × 8 area.
2.G.A.3Eliminate PossibilitiesOne straight cut in half shortens just one side by half — here 8 becomes 4 while the other side stays 8, so the piece is 4 by 8.
- Draw the square and its cut
- Halve one side, keep the other
- Read off and match the dimensions