AMC 8 · 2006 · #5

Grade 5 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Nested Frame / Concentric Difference

area-rectanglesarea-trianglesreflection-symmetrysimilar-figures area-differenceidentify-subproblems ↑ Prerequisites: area-rectanglesline-symmetry

📏 Short solution 💡 2 insights 📊 Diagram

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Problem

Inside a larger square, points A, B, C, D are the midpoints of its four sides. Joining them in order makes a smaller (tilted) square. The larger square has area 60. What is the area of the smaller square?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The picture is already given, but it pays off to add two more lines: the diagonals AC and BD of the smaller square. Those diagonals are horizontal and vertical, and they split the larger square into 8 small right triangles that are all congruent by the symmetry of midpoints. Once we see those 8 equal triangles, the answer comes from counting: 4 of them tile the smaller square and 4 tile the leftover corners, so the smaller square is exactly half of the larger. No algebra, no Pythagorean theorem.

1STEP 1

Draw the inner square's two diagonals AC and BD; they cross at the center of the larger square.

Add segments AC (vertical) and BD (horizontal) through the center

2STEP 2

These cuts plus the inner square's sides slice the larger square into 8 congruent right triangles.

Larger square = 8 congruent right triangles

3STEP 3

The two diagonals cut the inner square into 4 triangles; the other 4 fill the corners, so it is half the larger square.

Smaller square = 4 of the 8 triangles =

\frac{4}{8}

of the larger square =

\frac{1}{2}

4STEP 4

Half of 60 gives the inner square's area, 30, which is choice (D).

Area of smaller square =

\frac{1}{2}

× 60 = 30 → (D)

Answer

Cross-check with a side length. If the larger square has side s, then s² = 60. Each side of the smaller square is the hypotenuse of a right triangle with legs

\frac{s}{2}

and

\frac{s}{2}

, so its squared length is (

\frac{s}{2}

)² + (

\frac{s}{2}

)² =

\frac{s²}{2}

= 30. The area of the smaller square equals the square of its side, namely 30. That matches answer (D) and confirms the "half of 60" reasoning from the diagram. The answer also has to be less than 60 and clearly more than a quarter of it, which rules out (A), (B), and (E) on sight.

💡Key takeaway

When midpoints of a square's sides are joined, the inner square is always half the area of the outer one. Adding two diagonals to the picture makes that fact countable instead of computable.

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