AMC 10 · 2012 · #2
Grade 3 geometry-2dA square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?
Pick an answer.
AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A square measuring $8$ by $8$ is cut straight across into two matching (congruent) rectangles. Find the length and width of one of those rectangles.
Givens: The starting shape is a square with side length $8$; It is cut in half into two congruent rectangles; Answer choices: (A) $2$ by $4$, (B) $2$ by $6$, (C) $2$ by $8$, (D) $4$ by $4$, (E) $4$ by $8$
Unknowns: The two side lengths (dimensions) of one of the resulting rectangles
Understand
Restated: A square measuring $8$ by $8$ is cut straight across into two matching (congruent) rectangles. Find the length and width of one of those rectangles.
Givens: The starting shape is a square with side length $8$; It is cut in half into two congruent rectangles; Answer choices: (A) $2$ by $4$, (B) $2$ by $6$, (C) $2$ by $8$, (D) $4$ by $4$, (E) $4$ by $8$
Plan
Primary tool: #1 Draw a Diagram
Secondary: #17 Visualize Spatial Relationships, #3 Eliminate Possibilities
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.
Execute — Answer: E
3.G.A.2 Step 1 Draw the square and its cut
- Sketch the square, every side $8$.
- Cutting it in half means one straight cut through the middle, splitting it into two equal rectangles.
- That cut runs parallel to one pair of sides, so it slices across the square rather than shrinking it on all sides.
💡 A single straight cut can only shorten the shape in one direction, not both.
3.OA.C.7 Step 2 Halve one side, keep the other
- The cut lands on the middle of one pair of sides, splitting a side of length $8$ into two equal pieces: $8 \div 2 = 4$.
- The pair of sides running along the cut is untouched, so those still measure $8$.
- So each rectangle has one dimension $4$ and one dimension $8$.
💡 Cutting in half divides one length by $2$ while the crosswise length rides along unchanged.
2.G.A.3 Step 3 Read off and match the dimensions
- One rectangle is $4$ by $8$.
- Check the traps: $4$ by $4$ (D) would need both sides halved, and $2$ by $8$ (C) would need the side split into four, neither of which one straight half-cut does.
- The dimensions $4$ by $8$ match choice (E).
💡 Two equal shares of the square must together rebuild it, and $4 \times 8$ twice gives back the $8 \times 8$ area.
3.G.A.2 Sketch the square, every side $8$. Cutting it in half means one straight cut thr 3.OA.C.7 The cut lands on the middle of one pair of sides, splitting a side of length $8$ 2.G.A.3 One rectangle is $4$ by $8$. Check the traps: $4$ by $4$ (D) would need both sid Review
Reasonableness: Check by area: the whole square is $8 \times 8 = 64$, so each half should have area $32$. A $4$ by $8$ rectangle has area $4 \times 8 = 32$ — exactly half, which fits. The trap $4$ by $4$ has area $16$ (a quarter, too small) and $2$ by $8$ has area $16$ as well, so only $4$ by $8$ gives the correct half-area, confirming (E).
Alternative: Skip the picture and reason from area alone: each piece must have area $64 \div 2 = 32$, and it must still share a full side of length $8$ with the square (the uncut side). Then the other side is $32 \div 8 = 4$, giving $4$ by $8$, choice (E).
CCSS standards used (min grade 3)
3.G.A.2Partition shapes into equal parts with equal areas (Understanding that cutting the square in half produces two equal-area rectangles from one straight cut.)3.OA.C.7Fluently multiply and divide within 100 (Halving the cut side, $8 \div 2 = 4$, and checking areas like $4 \times 8 = 32$.)2.G.A.3Partition circles and rectangles into two, three, or four equal shares (Recognizing that two congruent halves reassemble into the original square, ruling out the trap dimensions.)
⭐ One 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 $8$.
⭐ One 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 $8$.
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