AMC 10 · 2021 · #9

Grade 8 arithmetic

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

coordinate-geometryrotation-isometryreflection-symmetrytransformations-composition work-backwardsidentify-subproblems ↑ Prerequisites: coordinate-geometryreflection-symmetry

📏 Medium solution 💡 3 insights

Problem

Start with an unknown point P(a, b). First rotate it counterclockwise by 90° about the point (1, 5); then reflect the result about the line y = -x. The final image is (-6, 3). Find b - a.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2021 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Work Backwards

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.

1STEP 1

Reflection over y = -x is its own inverse, so apply it again to (-6, 3), giving (-3, 6) — the point right after the rotation.

(-6, 3) → (-(3), -(-6)) = (-3, 6)

2STEP 2

Undo the rotation by first translating so C sits at the origin: (-3, 6) - (1, 5) = (-4, 1).

(-3, 6) - (1, 5) = (-4, 1)

3STEP 3

Rotate 90° clockwise about the origin with the rule (x, y) → (y, -x): (-4, 1) becomes (1, 4).

(-4, 1) → (1, -(-4)) = (1, 4)

4STEP 4

Translate back by adding C: (1, 4) + (1, 5) = (2, 9), so P = (a, b) with a = 2, b = 9.

(1, 4) + (1, 5) = (2, 9) = (a, b)

5STEP 5

Subtract: b - a = 9 - 2 = 7, which is choice (D).

b - a = 9 - 2 = 7 → (D)

Answer

Run the transformations forward on P = (2, 9) to confirm. Translate by -C: (2, 9) - (1, 5) = (1, 4). Rotate 90° CCW about origin (rule (x, y) → (-y, x)): (1, 4) → (-4, 1). Translate back by +C: (-4, 1) + (1, 5) = (-3, 6). Reflect over y = -x: (-3, 6) → (-6, 3). Final image (-6, 3) matches the given. b - a = 7 is the right answer.

💡Key takeaway

This AMC 10 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).

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