Identify the rotations and reflections that carry a figure onto itself and use symmetry to analyze properties of shapes

Learning goal: Identify the symmetries of a figure, meaning the rotations and reflections that carry it onto itself; identify figures with line symmetry and draw their lines of symmetry; identify figures with rotational symmetry and determine the angle of rotation; and use both kinds of symmetry to analyze the properties of rectangles, parallelograms, trapezoids and regular polygons.

What a symmetry is. A symmetry of a figure is a rigid motion that carries the figure onto itself: after the motion, the figure occupies exactly the same set of points as before, even though individual points may have moved. A line symmetry (also called reflection symmetry) is a reflection across a line that carries the figure onto itself; that line is a line of symmetry, and folding the figure along it makes the two halves coincide exactly. A rotational symmetry is a rotation, about some center and through some angle strictly between 0° and 360°, that carries the figure onto itself. For a figure with finitely many rotational symmetries, such as a polygon, the angle of rotational symmetry is the smallest such angle, and every multiple of it is also a symmetry; a circle is carried onto itself by every rotation about its center, so it has no smallest angle. The identity rotation of 0° (or 360°) carries every figure onto itself and is not counted as evidence of rotational symmetry. A figure "has rotational symmetry" only when some smaller angle works.

Testing for a line of symmetry. A proposed line is a line of symmetry only if every point of the figure reflects to a point of the figure. It is not enough that the figure looks balanced. The diagonal of a non-square rectangle is a classic trap: the rectangle with vertices (0, 0), (6, 0), (6, 4), (0, 4) looks balanced about its diagonal from (0, 0) to (6, 4), but reflecting the vertex (6, 0) across that diagonal gives a point off the rectangle, roughly (2.3, 5.5), well above the top side y = 4. The diagonal is not a line of symmetry. The two lines through the midpoints of opposite sides, x = 3 and y = 2, are: reflecting across x = 3 sends (0, 0) to (6, 0), (6, 0) to (0, 0), (6, 4) to (0, 4) and (0, 4) to (6, 4), so the vertex set, and with it the whole rectangle, is carried onto itself. A non-square rectangle has exactly 2 lines of symmetry.

Testing for rotational symmetry. For the same rectangle, rotate 180° about the center (3, 2). The rule for a half turn about (3, 2) is (x, y) → (6 − x, 4 − y). The vertex (0, 0) goes to (6, 4), (6, 0) to (0, 4), (6, 4) to (0, 0) and (0, 4) to (6, 0): each vertex lands on a vertex, so the rectangle is carried onto itself. Rotating 90° about the center does not work, because (0, 0) would go to (5, −1), which is outside the rectangle. A non-square rectangle has rotational symmetry with angle 180° and nothing smaller. A square gains the 90° rotation and two more lines of symmetry, its diagonals: for the square (0, 0), (4, 0), (4, 4), (0, 4) the diagonal y = x swaps (4, 0) with (0, 4) and fixes the other two vertices, so it is a line of symmetry, and the square has 4 lines of symmetry and rotational symmetries of 90°, 180° and 270°.

Parallelograms and trapezoids. A parallelogram with no right angle and unequal adjacent sides, such as (0, 0), (5, 0), (7, 3), (2, 3), has no line of symmetry at all. A reflection symmetry of a quadrilateral must send vertices to vertices, so its line either passes through two opposite vertices or swaps the vertices in pairs. A line through opposite vertices A and C would have to be the perpendicular bisector of BD, making AB = AD, but here the adjacent sides are 5 and √13. A line swapping adjacent pairs would be the perpendicular bisector of two opposite sides, forcing right angles at their ends, and there are none. A line swapping opposite vertices would be the perpendicular bisector of both diagonals, which is impossible because the diagonals cross. So no line works. It does have rotational symmetry of 180° about the point where its diagonals meet, here (3.5, 1.5): the half turn (x, y) → (7 − x, 3 − y) sends (0, 0) to (7, 3), (5, 0) to (2, 3), and back again. A rhombus that is not a square has 2 lines of symmetry, its diagonals, and 180° rotational symmetry. An isosceles trapezoid that is not a rectangle, with exactly one pair of parallel sides and the other pair congruent, has exactly 1 line of symmetry, the perpendicular bisector of the parallel sides, and no rotational symmetry, because a half turn would put the shorter parallel side where the longer one was. A trapezoid with no equal legs has neither kind of symmetry.

Regular polygons. A regular polygon with n sides has n lines of symmetry and rotational symmetry with angle 360° ÷ n. An equilateral triangle has 3 lines of symmetry, each through a vertex and the midpoint of the opposite side, and rotational symmetries of 120° and 240°. A regular hexagon has 6 lines of symmetry, 3 through opposite vertices and 3 through midpoints of opposite sides, and rotational symmetries of 60°, 120°, 180°, 240° and 300°. For an odd n every line passes through one vertex and the midpoint of the opposite side; for an even n half the lines pass through pairs of opposite vertices and half through pairs of opposite midpoints.

Using symmetry to analyze properties. Symmetries are more than a classification game: because a symmetry is a rigid motion, it preserves lengths and angles, so any two parts of a figure exchanged by a symmetry must be congruent. The half turn of a parallelogram about the meeting point of its diagonals exchanges the two pairs of opposite sides, which shows opposite sides of a parallelogram are congruent; it also carries each diagonal onto itself with its endpoints swapped, which shows the meeting point is the midpoint of both diagonals, so the diagonals bisect each other. Reflecting an isosceles trapezoid across its line of symmetry swaps the two legs and swaps the two base angles at each parallel side, showing the legs are congruent and the base angles are congruent in pairs. Reflecting a rectangle across the line x = 3 swaps the two diagonals, so the diagonals of a rectangle are congruent. Each of these is a genuine argument, provided the symmetry has first been verified.

The misconception. Students frequently claim a parallelogram has line symmetry "because both halves are the same". The two halves cut by a diagonal are congruent triangles, but they are related by a rotation, not a reflection; reflecting one half across the diagonal produces a kite shape, not the parallelogram. Congruent halves do not imply a line of symmetry. The test is always the same: reflect a vertex across the proposed line and see whether it lands on the figure.

Worked, rectangle: Rectangle with vertices (0, 0), (8, 0), (8, 2), (0, 2). Candidate lines: x = 4 sends (0, 0) to (8, 0) and (0, 2) to (8, 2), a symmetry; y = 1 sends (0, 0) to (0, 2) and (8, 0) to (8, 2), a symmetry; the diagonal from (0, 0) to (8, 2) sends (8, 0) to a point about (7.1, 3.8), above the top side y = 2 and off the figure, not a symmetry. Rotation of 180° about (4, 1) by (x, y) → (8 − x, 2 − y) sends every vertex to a vertex. Result: 2 lines of symmetry, rotational symmetry of angle 180°.

Worked, equilateral triangle: Vertices at (0, 0), (6, 0) and (3, 3√3), where 3√3 ≈ 5.20. The line x = 3 is the perpendicular bisector of the base and passes through the apex; reflecting swaps (0, 0) and (6, 0) and fixes the apex, so it is a line of symmetry. By the same reasoning the line through each vertex and the midpoint of the opposite side is a line of symmetry: 3 lines. The center is at (3, √3) ≈ (3, 1.73), one third of the height above the base. Rotating 120° about the center sends each vertex to the next, so 120° and 240° are rotational symmetries and 120° is the angle of rotational symmetry.

Worked, isosceles trapezoid: Vertices (0, 0), (10, 0), (8, 4), (2, 4). The parallel sides are y = 0 (length 10) and y = 4 (length 6). Reflecting across x = 5: (0, 0) ↔ (10, 0) and (2, 4) ↔ (8, 4), a line of symmetry. The legs are congruent because the reflection exchanges them: both have run 2 and rise 4, length √20. A half turn about the center would send the side of length 10 to where the side of length 6 is, so there is no rotational symmetry. Result: 1 line of symmetry, no rotational symmetry.

Worked, regular hexagon analysis: A regular hexagon has 6 lines of symmetry and rotational symmetry of angle 60°. Rotating by 60° about the center carries each vertex to the next, so all six sides are congruent (each is carried onto the next) and all six interior angles are congruent. Rotating by 180° carries each side to the opposite side, so opposite sides are parallel, because a half turn takes every line to a parallel line. Both facts follow from the symmetry without measuring anything.

Recording task: Draw on grid paper the non-square rectangle (0, 0), (6, 0), (6, 4), (0, 4), the parallelogram (0, 0), (5, 0), (7, 3), (2, 3), the isosceles trapezoid (0, 0), (10, 0), (8, 4), (2, 4) and a regular hexagon. For each figure draw every line of symmetry you claim, and record a table with columns "figure", "number of lines of symmetry", "angle of rotational symmetry (or none)", "one property the symmetry proves". Expected entries: rectangle 2 lines, 180°; parallelogram 0 lines, 180°; isosceles trapezoid 1 line, none; regular hexagon 6 lines, 60°. For one claimed line of symmetry, show the coordinates of a reflected vertex landing on the figure, and for one rejected line show a reflected vertex that misses.

Independent performance: Choose a regular polygon with 5, 8 or 12 sides and a quadrilateral that is not one of the standard names (a kite, or a figure you draw yourself that has at least one line of symmetry or a rotational symmetry of less than 360°, since a figure with no symmetry gives nothing to use). For each, determine every line of symmetry and the angle of rotational symmetry, and then use one symmetry to prove a property of the figure in a short paragraph. Success criteria: the number of lines of symmetry and the angle of rotation are correct and justified by tracing paper, folding, coordinates or software; each line of symmetry is drawn in the correct position (through vertices or midpoints, as appropriate); the property you prove is stated precisely; the argument names the symmetry used and the fact that rigid motions preserve length and angle; and one sentence explains why congruent halves alone do not establish a line of symmetry.