AMC 10 · 2004 · #9

Grade 8 geometry-2d
Archetype Paper Fold-and-Cut Symmetry └ Sub-archetype Composition of Reflections of a Pictured Figure
rotation-isometryreflection-symmetrytransformations-composition physical-representationspatial-visualization ↑ Prerequisites: coordinate-geometry
📏 Long solution 💡 2 insights
Problem
The point negative three, two is turned 90 degrees clockwise about the origin to reach B. Then B is reflected across the line where the coordinates are equal to reach C. Give the coordinates of C.

Pick an answer.

(A)
(-3,-2)
(B)
(-2,-3)
(C)
(2,-3)
(D)
(2,3)
(E)
(3,2)
How to solve
Strategy Draw a Diagram

The problem is about where a point lands after two moves, the signature setting for Tool #1 (Draw a Diagram): plot the point, and read each move off the picture rather than off a half-remembered formula. Tool #17 (Visualize Spatial Relationships) supplies the checks that catch a turn made in the wrong direction — same distance from the origin, a right angle at the origin, and the correct quadrant. Tool #15 (Organize Information in More Ways) then rewrites both moves as coordinate rules and chains them into a single rule, an independent computation that never mentions B. Tool #3 (Eliminate Possibilities) closes by showing why every choice survives a length check, so only direction and order can decide.

1STEP 1

Pin down the clockwise rule

Testing the turn on east and north pins the clockwise rule without memorizing.

(1,0)↦(0,-1), (0,1)↦(1,0) ⟹ (x,y)↦(y,-x)
2STEP 2

Turn the point, then check it

Applying it lands B in the first quadrant, confirmed by length and angle.

B=(2, -(-3))=(2,3); √((-3)²+2²)=√(2²+3²)=√(13); (-3)(2)+(2)(3)=0
3STEP 3

Reflect across the diagonal

Reflecting across the diagonal simply swaps the coordinates.

midpoint=((a+b)/2,(a+b)/2)∈{x=y}, (b-a, a-b)·(1,1)=0 ⟹ (a,b)↦(b,a); (2,3)↦(3,2)
4STEP 4

Chain the two rules into one

Composing the two rules is just a flip across the vertical axis, giving 3, 2.

(x,y) → (y,-x) → (-x,y); (-3,2)↦(3,2)
5STEP 5

Rule out the traps

Every trap choice fails direction or order, so C is 3, 2, choice (E).

|(-3,-2)|=|(-2,-3)|=|(2,-3)|=|(2,3)|=|(3,2)|=√(13) ⟹ direction decides; C=(3,2) (E)
Answer
(3,2)
The two routes agree: stepping through B=(2,3) and applying the composed rule (x,y)↦(-x,y) both land on (3,2), and (3,2) is √(13) from the origin, the distance both moves must preserve. A structural check backs this up: a rotation preserves orientation and a reflection reverses it, so doing one of each must reverse orientation overall — and the composed map (x,y)↦(-x,y), reflection across the y-axis, does exactly that. Notice also that C=(3,2) is the plain mirror image of the starting point (-3,2) across the y-axis, which is the composed rule stated in words. Since all five choices share the distance √(13), a student who checks only length learns nothing here; the problem is built so that only the direction of the turn and the order of the moves separate right from wrong.
💡Key takeaway

Write each move as a rule for what happens to (x,y), then chain the rules together — the chain gives the answer and checks it at the same time.

  • Pin down the clockwise rule
  • Turn the point, then check it
  • Reflect across the diagonal
  • Chain the two rules into one
  • Rule out the traps

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