AMC 10 · 2021 · #5
Grade 8 geometry-2dPick an answer.
Tool #11 (Work Backwards): the problem hands us the end state and asks for the start — undo the reflection first, then undo the rotation. Tool #7 (Subproblems): each transformation is its own self-contained inverse. Tool #1 (Diagram): sketch the four key points (final image, post-rotation image, rotation center C, and P) on a quick coordinate grid to keep signs straight. Tool #3 (Eliminate): the answer is one of {1, 3, 5, 7, 9} — all odd — so a parity sanity-check (if a, b are integers, b - a inherits the right parity) is built in.
Undo the reflection
Undo the last transformation first.
Grade 8 reflections: reflecting twice over the same line returns the original.
Reflecting twice over the same line returns the figure to exactly where it started.
▸ Why?
A reflection moves the figure without stretching, so nothing is lost either time.
▸ Why?
The second reflection undoes the first exactly, so the pair cancels out.
Shift to the centre
Shift so the centre becomes the origin.
Grade 8 translation: shift the center to the origin so a simple rotation rule applies.
8.G.A.3Identify SubproblemsUndo the rotation
Turn ninety degrees the other way.
Grade 8 rotation: 90° CW sends (x, y) → (y, -x).
8.G.A.1Work BackwardsShift back
Shift back to where it was.
Grade 8 translation: undo the shift to land back in the original coordinate system.
8.G.A.3Identify SubproblemsTake the difference
The difference is 7.
Grade 4 subtraction: 9 - 2 = 7, exactly choice (D).
4.NBT.B.4Eliminate PossibilitiesThis AMC 12 problem only needs Grade 8 transformation rules you already know — reflect over y = -x (a self-inverse) to undo step 2, then translate to center, 90° CW rotate, translate back to undo step 1. Starting from (-6, 3), the chain gives P = (2, 9), so b - a = 9 - 2 = 7, choice (D).