AMC 10 · 2003 · #9
Grade 8 geometry-2dPick an answer.
The four symmetries are geometric moves, so tool #1 (Draw a Diagram) turns them into something you can see: plot (2,3), then plot every mirror image the rules demand. Tool #2 (Make a Systematic List) keeps that generation honest, because the question is a count and a missed point or a double-counted point ruins it. Tool #14 (Extreme Principle) names the real job: the word "smallest" means two separate claims have to be settled, a floor (every legal S must contain at least these points) and a ceiling (this particular set of points is itself legal). Getting the forced points is only half the problem; showing nothing more is forced is the other half.
Write each symmetry as a coordinate rule
Each move is a coordinate rule; the half-turn is just the two axis flips combined.
A symmetry is a rule that sends every member of the set to another member, so writing it in coordinates turns geometry into arithmetic you can actually run.
8.G.A.3Draw A DiagramChase the point through the rules
Chasing the given point through the rules forces all of (±2,±3) and (±3,±2).
Symmetry rules are one-way demands: once a point is in, its mirror images have no choice but to join it.
6.NS.C.8Make A Systematic ListConfirm the eight points are all different
Since 2 and 3 differ and neither is zero, those are eight distinct points.
A count of generated images is trustworthy only when nothing on the list collided, which is exactly what unequal and nonzero coordinates guarantee.
6.NS.C.7Make A Systematic ListCheck the eight points close up
Testing every rule on that set produces nothing new, so it is genuinely closed.
A set stops growing exactly when every rule sends it back into itself, and that is the moment you know you have the smallest one.
8.G.A.1Extreme PrincipleCount and conclude
Bound and construction meet, so the smallest possible size is 8, choice (D).
When independent choices build each object exactly once, multiplying the number of choices counts them all.
Each of the eight points is built by choosing a sign for each coordinate, so the choices multiply.
▸ Why?
One sign choice puts no limit on the other, so every combination of signs really occurs.
▸ Why?
Each symmetry is a flip or a turn, and such a move lands a point of the set onto another point of the set without changing distances.
Apply every symmetry rule to the point you were given until nothing new appears, then check that the list you ended with obeys all the rules itself, and that list is the smallest set possible.
- Write each symmetry as a coordinate rule
- Chase the point through the rules
- Confirm the eight points are all different
- Check the eight points close up
- Count and conclude