AMC 8 · 2005 · #13

Grade 3 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Rectilinear Frame / Box Border

area-rectanglesarea-differenceperimeter area-differenceidentify-subproblems ↑ Prerequisites: area-rectangles

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Hexagon ABCDEF is an L-shape with every corner a right angle. Going around in order, AB = 8 (top), BC = 9 (right side), and FA = 5 (left side). The polygon has area 52. Find DE + EF.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

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.

1STEP 1

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.

Rectangle ABCO: width = AB = 8, height = BC = 9

2STEP 2

On the left, FA stacked on DE must span the full height BC, so FA + DE = BC gives DE = 4.

FA + DE = BC → 5 + DE = 9 → DE = 4

3STEP 3

The L's area is the big rectangle minus the DE-by-EF cut-out, so 72 − 4 × EF = 52 gives EF = 5.

52 = 8 × 9 - EF × DE = 72 - 4 EF → 4 EF = 20 → EF = 5

4STEP 4

Add the two subproblem answers for the requested sum: DE + EF = 4 + 5.

DE + EF = 4 + 5 = 9 → (C)

Answer

Plug the numbers back into the picture. With DE = 4 and EF = 5, the cut-out corner is a 4 × 5 rectangle of area 20. The big rectangle has area 8 × 9 = 72. Subtracting, 72 - 20 = 52, which matches the given area exactly. Magnitude check: DE must be smaller than BC = 9 (it is only part of the left side), and EF must be smaller than AB = 8 (it is only part of the top-to-bottom step), so DE + EF ＜ 9 + 8 = 17 — answer 9 comfortably fits. Choice (A) 7 would force 4 EF = 12, giving area 72 - 12 = 60 ≠ 52; (E) 11 would force 4 EF = 28, giving area 72 - 28 = 44 ≠ 52. Only 9 works.

💡Key takeaway

Close 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.

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