AMC 10 · 2016 · #16

Grade 8 geometry-2d
Archetype Paper Fold-and-Cut Symmetry └ Sub-archetype Composition of Reflections of a Pictured Figure
transformations-compositionrotation-isometryreflection-symmetry work-backwards ↑ Prerequisites: transformations-composition
📏 Long solution 💡 3 insights
Problem
Triangle ABC has vertices A(0,2), B(-3,2), C(-3,0). It is first reflected across the x-axis to give △ A'B'C', then that image is rotated 90^° counterclockwise about the origin to give △ A''B''C''. Which single transformation carries △ A''B''C'' back onto the original △ ABC?

Pick an answer.

(A)
counterclockwise rotation about the origin by 90°
(B)
clockwise rotation about the origin by 90°
(C)
reflection about the x-axis
(D)
reflection about the line y = x
(E)
reflection about the y-axis

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Visualize Spatial Relationships

The question is about how reflections and rotations move points, so Tool #17 (Visualize Spatial Relationships) leads: picture the flip and the turn acting on the plane. Instead of dragging all three vertices through both moves, Tool #4 (Introduce a Variable) tracks one general point (x,y) and finds the coordinate rule for the whole two-step process. The phrase 'return to △ ABC' is a Tool #11 (Work Backwards) cue: we need the move that undoes the combined map. Tool #3 (Eliminate Possibilities) then confirms the winner against the five choices.

1STEP 1

Track one general point

Track one general point (x,y): every vertex obeys the same rule, so one point captures the whole transformation.

P=(x,y)
2STEP 2

Reflect across the x-axis

Reflecting across the x-axis keeps the horizontal coordinate and flips the vertical sign, so (x,y) lands at (x,-y).

(x,y) → (x,-y)
3STEP 3

Rotate 90 degrees counterclockwise

A 90^° counterclockwise turn sends (a,b) to (-b,a); applying it to (x,-y) gives (y,x).

(x,-y) → {90^° ccw} (-(-y),x)=(y,x)
4STEP 4

Name the combined move

The two moves together send (x,y) to (y,x)—swapping the coordinates, exactly the mirror across the line y=x.

(x,y) ⟶ (y,x) = reflection across y=x
5STEP 5

Undo it to get back

A reflection is its own inverse, so mirroring across that same line once more sends (y,x) back to (x,y). The answer is (D).

(y,x) → (x,y) → (D)
Answer
reflection about the line y = x
Check the rule on a real vertex. A(0,2) reflects to (0,-2), then rotates to (2,0)=A''; reflecting A''(2,0) across y=x gives (0,2)=A, exactly the original. The other choices fail: a 90^° counterclockwise rotation sends B''(2,-3) to (3,2), not B(-3,2); reflection across the x-axis sends A''(2,0) to (2,0), not A(0,2); reflection across the y-axis sends A''(2,0) to (-2,0), not A(0,2). Only choice (D) works for all vertices.
💡Key takeaway

Follow one general point (x,y): the flip then the turn sends it to (y,x), which is just the mirror across y=x, and reflecting across that same line again sends it home, choice (D).

  • Track one general point
  • Reflect across the x-axis
  • Rotate 90 degrees counterclockwise
  • Name the combined move
  • Undo it to get back

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