AMC 10 · 2016 · #9

Grade 9 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
coordinate-geometrypythagorean-theoremmidpoint-formulalinear-equations-one-var convert-to-algebracoordinate-geometry ↑ Prerequisites: pythagorean-theoremmidpoint-formula
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
Four squares sit in the corners of a big square and a fifth sits turned in the middle. Report the parts of their common side.

Pick an answer.

(A)
7
(B)
8
(C)
9
(D)
10
(E)
11
How to solve
Strategy Introduce a Variable

Every length in the picture is controlled by one number, so tool #4 (Introduce a Variable) is the whole game: call the common side length s and write every other length in terms of it. Tool #1 (Draw a Diagram) turns the picture into coordinates, which converts the vague phrase "the midpoint lands on a vertex" into two points that must be equal. Tool #17 (Visualize Spatial Relationships) supplies the one fact the tilt hides: the middle square is the same small square turned 45°, so its centre-to-vertex distance is half of s√(2). Matching the midpoint to the corner square's inner vertex then gives a single linear equation in s.

1STEP 1

Put the picture on coordinates

Coordinates name every vertex.

inner vertex = (1-s, 1-s), tilted vertices = (1/2,1/2± r), (1/2± r,1/2)
2STEP 2

Half-diagonal of the tilted square

The turned square's half-diagonal follows at once.

(diagonal)² = s² + s² → diagonal = s√(2), r = s√(2)/2
3STEP 3

Turn the midpoint clue into an equation

The midpoint clue becomes one equation.

1/2+r/2 = 1-s ⟺ r = 1-2s
4STEP 4

Solve the linear equation for s

It is linear, so it solves at once.

s√(2)/2 = 1-2s → s√(2) = 2-4s → s(√(2)+4) = 2 → s = 2/(4+√(2))
5STEP 5

Rewrite in the requested form

Rationalising gives 11, choice (E).

s = 2/(4+√(2))·(4-√(2))/(4-√(2)) = (2(4-√(2)))/14 = (4-√(2))/7 → a+b = 11 → (E)
Answer
11
Put numbers back into the picture. With √(2)≈ 1.414, the side is s = (4-√(2))/7 ≈ 0.369. Two corner squares along one side of the unit square take up 2s ≈ 0.739, leaving a gap of 1-2s ≈ 0.261 in the middle of that side, and the tilted square's half-diagonal is s√(2)/2 ≈ 0.261 — it reaches exactly into that gap, which is what the picture shows. The five squares cover 5s² ≈ 0.68 of the unit square's area, safely under 1, so disjoint interiors are possible. Since a=4 and b=7 are positive integers, a+b=11, which is choice (E).
💡Key takeaway

Call the common side s: two corner squares leave a gap of 1-2s along each side of the big square, the tilted square's half-diagonal s√(2)/2 has to fill exactly that gap, and solving gives s=(4-√(2))/7, so a+b=11.

  • Put the picture on coordinates
  • Half-diagonal of the tilted square
  • Turn the midpoint clue into an equation
  • Solve the linear equation for s
  • Rewrite in the requested form

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