Find the complete symmetry set

Try

Cut out a rectangle whose sides measure 6 and 2 units. Turn it halfway around its center: its outline returns to its original position. Turn it one quarter-turn: the long and short sides exchange places, and the outline does not coincide. A symmetry is a rigid motion mapping an entire figure onto itself. It need not fix each individual point.

Now fold across a line joining the midpoints of opposite sides. Both halves match. Folding across a diagonal behaves differently: equal-looking corners are not enough when the adjacent side lengths differ. The question is about the complete figure and exact distances, not a decorative visual resemblance.

Predict

Predict which of these claims survive: every parallelogram has a half-turn symmetry; every parallelogram has a reflection symmetry; every rectangle has diagonal reflection symmetry; every regular pentagon has a reflection through each vertex and the opposite side midpoint. Give a proposed axis or center for each claim before checking it.

Experiment

The diagram uses equal horizontal and vertical units. Trace the rectangle and the slanted parallelogram separately. For the rectangle, test its two midpoint lines. For the parallelogram, rotate each labeled vertex through a half-turn about its diagonal midpoint.

For the parallelogram, the coordinate rule is (x,y)↦(5−x,2−y). It exchanges (0,0) with (5,2) and (4,0) with (1,2), and maps all connecting edges accordingly. Its adjacent side lengths are 4 and √5, and the adjacent sides are not perpendicular. Test both its diagonals and lines through opposite side midpoints as proposed reflection axes; none works.

Understand

For a rectangle with unequal sides, the two midpoint axes and the half-turn preserve its outline. There are no other symmetries: a symmetry must send a vertex to a vertex and preserve the lengths of its two incident sides. Each chosen image vertex determines where the long-side neighbor goes, leaving exactly four choices in total, including identity. The two reflections and the two rotations already realize all four.

A general parallelogram has a half-turn because its opposite vertices have the same midpoint. The coordinate proof above exemplifies the general rule C=A+u+v, B=A+u, D=A+v: the map X↦A+C−X exchanges opposite vertices and sides. Reflection is an additional property, not automatic. A reflection of a quadrilateral either fixes two opposite vertices and exchanges the others, or exchanges two pairs of adjacent vertices. The first pattern forces equal adjacent sides, making a parallelogram a rhombus; the second pattern forces its adjacent directions perpendicular, making it a rectangle. Our slanted example satisfies neither condition. A square satisfies both and has four reflection axes.

A regular n-gon, with n≥3, has n rotations through multiples of 360/n degrees and n reflections. For odd n each reflection axis passes through a vertex and the opposite edge midpoint. For even n, half the axes join opposite vertices and half join opposite edge midpoints. To prove there are no more, choose the image of one vertex in n ways and the image of an adjacent vertex in two ways. Adjacency is preserved, and these choices determine the boundary order and the rigid map on the noncollinear polygon. The 2n known motions exhaust those possibilities.

Apply

Compare an isosceles trapezoid with unequal bases and a nonsymmetric trapezoid. Here “trapezoid” means exactly one parallel side pair. The isosceles example has one perpendicular midpoint axis and no nonidentity rotation. A generic example may have no reflection at all. Draw coordinates and verify the axis maps every vertex and edge; never infer an exact symmetry from an approximate hand sketch.

For a regular triangle, list the rotations including identity and draw all reflection axes. Explain why a rotation angle smaller than 120 degrees cannot take a vertex to the next available vertex. This combines a constructive list with an exclusion argument.

Continue

Next construct motion witnesses between two different labeled figures. A symmetry is the special case where source and target are the same set. In either setting, the correspondence, full figure and distance-preserving nature of the steps must all be checked.