Competition · AMC preparation · step 4 of 4
AMC 8 · 2005 · #13
Grade 3 geometry-2d
Pick an answer.
AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #1 (Draw a Diagram) is the cracker: sketch the L-shape with the labels, then extend AF down and DC up until they meet at a new corner O. The L is now the big rectangle ABCO with a smaller rectangle cut out of one corner — and that cut-out rectangle has sides DE and EF. Tool #7 (Identify Subproblems) then breaks the question into two small pieces: (a) use the right-angle structure to find DE from the vertical sides, and (b) use the area equation "big rectangle minus cut-out = 52" to find EF. Choosing Tool #1 + Tool #7 over Tool #13 (Convert to Algebra) keeps the work at one short subtraction and one short equation — no system of equations needed.
Complete the rectangle
Extend FA down and DC up to a new corner O, forming rectangle ABCO — the L is that big rectangle minus a small cut-out corner FEDO.
Closing the L into a rectangle turns an unfamiliar 6-sided shape into a difference of two rectangles — shapes whose area is just length times width.
3.G.A.1Draw A DiagramFind DE from the sides
On the left, FA stacked on DE must span the full height BC, so FA + DE = BC gives DE = 4.
In any axis-aligned shape, the verticals on the left must sum to the same total as the verticals on the right — that is just "two paths between the same two horizontals have equal length."
3.MD.D.8Identify SubproblemsFind EF from the area
The L's area is the big rectangle minus the DE-by-EF cut-out, so 72 − 4 × EF = 52 gives EF = 5.
"Big rectangle minus small rectangle" is the cleanest way to handle any L-shape area.
The area of the L-shape equals the area of the big rectangle ABCO minus the area of the small cut-out rectangle FEDO, and that is what pins down EF.
▸ Why?
The big rectangle ABCO is exactly the L-shape and the small cut-out rectangle FEDO put together with no gap and no overlap, so the big area is those two areas added.
▸ Why?
Extending FA and DC to the new corner O fills the missing bite, so the L and that bite tile the whole big rectangle with nothing left over and nothing counted twice.
▸ Why?
Since big rectangle = L + cut-out, taking the cut-out area back off the big area leaves exactly the L's area, so one subtraction isolates the piece we want.
▸ Why?
Each rectangle's area is its width times its height, so the big one measures 8 × 9 and the cut-out measures EF × DE.
▸ Why?
A rectangle is filled by equal rows of unit squares — one row for each unit of height, each row holding a width of squares — so counting the squares is width times height.
Add the two pieces
Add the two subproblem answers for the requested sum: DE + EF = 4 + 5.
Combine the two subproblem answers — the last step of any split-it-up plan.
3.OA.D.8Identify SubproblemsClose the L-shape into a full rectangle, and the missing corner is a smaller rectangle. "Big rectangle minus cut-out = 52" gives EF, and "left verticals add up to the right vertical" gives DE — pure Grade 3 area arithmetic.
- Complete the rectangle
- Find DE from the sides
- Find EF from the area
- Add the two pieces
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