Competition · AMC preparation · step 4 of 4

AMC 8 · 2015 · #25

Grade 6 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation

area-rectanglesarea-trianglesspatial-visualizationreflection-symmetry identify-subproblemsarea-difference ↑ Prerequisites: area-rectanglesarea-triangles

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A 5 × 5 square has a 1 × 1 square cut out of each of its four corners, leaving a plus-shaped region. We want the area of the largest square that fits inside that plus shape.

Pick an answer.

(A)

(B)

$12+4\sqrt{2}$

(C)

(D)

$9+4\sqrt{5}$

(E)

AMC 8 2015 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 move that turns this from a scary AMC 8 #25 into a one-line area calculation: when you sketch the plus shape and the biggest tilted square inside it, you can see that the inscribed square's four corners must sit at the four inner corners of the cut-out notches — the only way to push outward as far as possible without hitting a removed unit square. Once the picture is right, Tool #7 (Identify Subproblems) splits the inscribed square into an upright inner 3 × 3 square (the part you can color in by eye) plus 4 congruent right triangles that fill the sides. Adding two simple areas gives the answer; no algebra and no Pythagoras needed.

1STEP 1

Sketch the plus shape

Sketch the plus on grid paper: erasing the four corner units leaves an upright inner square of side 3.

Inner upright square has side 5 - 1 - 1 = 3.

2STEP 2

Draw the tilted square

Draw the biggest tilted square inside the plus; its four vertices land on the notch corners, the midpoints of the plus's long edges.

Vertices of the inscribed square: (1,5), (5,4), (4,0), (0,1).

3STEP 3

Split into square and triangles

Split the tilted square into the upright 3 × 3 inner square plus 4 congruent right triangles, each with base 3 and height 1.

Tilted square area = (inner 3 × 3 square) + 4 × (notch triangle).

Decomposing a tilted polygon into an upright rectangle plus right triangles is the Grade 6 area-by-decomposition strategy.

The largest square inside the plus shape has the same area as an upright 3-by-3 square at its center plus four equal right triangles, one along each side, where each triangle stands on legs of length 3 and 1 and covers exactly half of the 3-by-1 rectangle on those legs.

▸ Why?

The central square and the four triangles fit inside the big tilted square with no overlaps and no gaps, tiling it completely, so their areas add up to the big square's area.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

▸ Why?

Each triangle rests on a full side of the central square, so its long leg is 3.

▸ Why?

That side runs along the original 5-unit edge with a 1-unit notch removed at each end, so it measures 5 minus 1 minus 1, which is 3.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

▸ Why?

The four triangles are all congruent, so each side of the square contributes the very same area.

▸ Why?

A quarter turn about the center of the plus shape lands it exactly on itself and carries each triangle onto the next, and turning a shape never changes its size.

Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).

▸ Why?

Each triangle covers exactly half of the 3-by-1 rectangle standing on its two legs.

▸ Why?

A copy of the triangle, given a half turn about the midpoint of its slanted side, matches the original exactly, since turning never changes size or shape.

Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).

▸ Why?

The triangle and that matching copy meet along the slanted side and together fill the 3-by-1 rectangle with no gap or overlap, so the triangle is one of two equal halves.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

6.G.A.1Identify Subproblems

4STEP 4

Compute the inner square area

The upright inner square has area 3 × 3 = 9.

3 × 3 = 9.

5STEP 5

Compute the four triangle areas

Each right triangle has legs 3 and 1, so area $\frac{1}{2}$ · 3 · 1 = $\frac{3}{2}$ ; the four together make 6.

4 × 1/2 · 3 · 1 = 4 × 3/2 = 6.

6STEP 6

Add the pieces

Add the pieces: 9 + 6 = 15, choice (C).

Largest inscribed square area = 9 + 6 = 15 → (C).

Answer

The original 5 × 5 square has area 25. The plus-shaped region has area 25 - 4 = 21, which equals answer (E). The largest square inside the plus must have area less than 21 but clearly more than 9 (the upright inner square is already 9 and we still have room to tilt), so the answer should sit strictly between 9 and 21. The four candidate values in that range are 12 + 4√(2) ≈ 17.66, 15, and 9 + 4√(5) ≈ 17.94. Our decomposition gives exactly 15, the smallest of the three, which matches the fact that any tilted square still has to dodge the four unit-square corners.

💡Key takeaway

Draw the picture first — the tilted square breaks into one 3 × 3 square plus four little right triangles, and Grade 6 area-by-pieces does the rest.

Sketch the plus shape
Draw the tilted square
Split into square and triangles
Compute the inner square area
Compute the four triangle areas
Add the pieces

A parent dashboard for the family lives at sensimlab.com.

Sketch the plus shape

Draw the tilted square

Split into square and triangles

Compute the inner square area

Compute the four triangle areas

Add the pieces

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