AMC 10 · 2003 · #14

Grade 8 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation
equilateral-trianglerotation-isometryspatial-visualizationcircumradius symmetry-argumentidentify-subproblems ↑ Prerequisites: equilateral-triangle
📏 Long solution 💡 3 insights 📊 Diagram
Problem
Square ABCD has area 16. On each side an equilateral triangle is built pointing away from the square. The four outer tips are K, L, M, N. Find the area of quadrilateral KLMN.

Pick an answer.

(A)
32
(B)
$16+16\sqrt{3}$
(C)
48
(D)
$32+16\sqrt{3}$
(E)
64
How to solve
Strategy Visualize Spatial Relationships

The tempting move is to say "KLMN looks like a square" and start computing. That is exactly the step that has to be earned, because everything after it depends on it. Tool #17 (Visualize Spatial Relationships) earns it in one stroke: turn the entire picture 90° about the center of ABCD and watch it land on itself, sending each tip to the next tip. A figure carried onto itself by a quarter turn about a point has its four tips equally far from that point and evenly spaced, so it is a square. Tool #1 (Draw a Diagram) supplies the one measurement that turn does not give — how far a tip sits from the center. Tool #7 (Identify Subproblems) splits the job into two small ones: find the diagonal, then convert a diagonal into an area.

1STEP 1

Side of the square is 4

Area 16 gives side 4, so every triangle has all sides 4.

s²=16 → s=4
2STEP 2

A quarter turn maps the picture to itself

A quarter turn about the centre sends each tip to the next, so KLMN is itself a square.

rotate 90° about O: K↦ L↦ M↦ N↦ K
3STEP 3

Walk from the center out to a tip

The triangle's height is 2√(3), so half the diagonal is 2+2√(3) and KM = 4+4√(3).

h=√(4²-2²)=2√(3), OK=2+2√(3), KM=4+4√(3)
4STEP 4

A square's area from its diagonal

A square's area from its diagonal is d² over 2.

[KLMN]=2·1/2 · d·d/2=d²/2
5STEP 5

Substitute and finish

Substituting gives 32+16√(3), choice (D).

[KLMN]=((4+4√(3))²)/2=(64+32√(3))/2=32+16√(3) → (D)
Answer
32+16√(3)
Rebuild KLMN from pieces and the total should match exactly. KLMN is made of the square (16), the four equilateral triangles (each √(3)/4 · 4²=4√(3), so 16√(3) in all), and the four corner gaps such as triangle NAK. In triangle NAK the sides AN and AK are both 4 and the angle between them is 360°-90°-60°-60°=150°, so the height from K onto line AN is 4sin 150°=2 and its area is 1/2 · 4 · 2=4; four of those give 16. The total is 16+16√(3)+16=32+16√(3), matching. Numerically that is about 59.7, comfortably bigger than the square plus triangles alone (≈ 43.7) and smaller than 64. Note that choice (B), 16+16√(3), is exactly what you get if you forget the four corner gaps.
💡Key takeaway

One quarter turn of the whole picture sends each tip to the next tip, so KLMN has to be a square; its diagonal is the square's side plus two triangle heights, 4+4√(3), and half of the diagonal squared is the area.

  • Side of the square is 4
  • A quarter turn maps the picture to itself
  • Walk from the center out to a tip
  • A square's area from its diagonal
  • Substitute and finish

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