AMC 10 · 2003 · #10
Grade 8 counting
Pick an answer.
The picture makes the five triangle positions look different, but the pentagon's own symmetry makes them interchangeable, so the honest way to count is to list every choice and then decide which listed choices are secretly the same figure. Tool #2 (Make a Systematic List) produces all ten two-side choices with nothing missed and nothing doubled. Tool #1 (Draw a Diagram) turns the pentagon into five numbered slots around a ring, which is the whole picture the problem depends on. Tool #5 (Look for a Pattern) finds the feature that never changes when the pentagon is turned or flipped — how far apart the two chosen sides are around the ring — and sorting by that feature is what collapses ten choices into a much smaller number of shapes. The final and most-skipped move is proving the surviving families really are different, not just different-looking.
Number the five slots around the ring
Number the slots around a ring, so a figure is just a pair of slots.
A regular pentagon looks the same after a fifth of a turn or a flip, so the slot numbers are only bookkeeping, not real differences.
4.G.A.3Draw A DiagramList all ten choices
There are 10 such pairs, but that counts choices, not shapes.
Listing every choice first is safe; deciding which ones coincide is a separate question that should not be guessed at.
7.SP.C.8Make A Systematic ListSort by the gap, and watch the list collapse
Turning and flipping cannot change the gap, and on a five-ring only two gaps exist.
Turning the pentagon slides both chosen slots by the same amount, so the gap between them is the only thing that survives.
Turning the pentagon slides both chosen slots by the same amount, so only the gap between them survives.
▸ Why?
A turn lays the pentagon onto a copy of itself, so a rotated choice is the same choice seen from a new angle.
▸ Why?
Moving both slots by the same step leaves the distance between them untouched, so that distance is what tells the cases apart.
Show the two families really are different shapes
One gap makes the triangles touch and the other keeps them apart, so the two are genuinely different, choice (B).
Rigid motions preserve which parts touch, so touching versus not touching is a difference no turn or flip can erase.
8.G.A.2Draw A DiagramWhen a shape can be turned and flipped, count the features that survive those moves — here only whether the two triangles touch — instead of counting the positions you happened to label.
- Number the five slots around the ring
- List all ten choices
- Sort by the gap, and watch the list collapse
- Show the two families really are different shapes