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 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 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 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).

4STEP 4

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

3 × 3 = 9.

5STEP 5

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

4 ×

\frac{1}{2}

· 3 · 1 = 4 ×

\frac{3}{2}

= 6.

6STEP 6

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.

More like this