1.9 Symmetry of a Figure — Transformations

Standards: G-CO.A.3 · G-CO.B.6

Decide whether two figures are congruent by identifying a sequence of rigid motions.

Describe the rotations and reflections that carry a figure onto itself.

Use the vocabulary in [missing reference: preimage-image-vocab] and the decision rule in [missing reference: rigid-motion-congruence-rule].

Definitions and rules

For rotational symmetry, identify the center, find the smallest positive turn that produces a match, then list its multiples less than 360°.

For reflection symmetry, test a candidate line by pairing every point with a mirror point the same perpendicular distance from the line.

A correct symmetry description names transformations, not visual resemblance.

Symmetry is a rigid motion back onto the same figure

A figure has a reflection or rotation symmetry when that transformation maps every point of the figure onto the figure itself.

List the reflection lines and all nontrivial rotation angles that produce a full match.

Why it has to be this: A partial match is insufficient because a symmetry must preserve the entire set of points in the figure.

Language for deciding congruence and describing symmetry.
Term	Meaning	Decision test	Nonexample
Congruent figures	Figures with the same size and shape.	A sequence of translations, rotations, and reflections maps one figure onto the other.	A dilation with scale factor 2 changes every nonzero length.
Rigid motion	A transformation that preserves every distance and angle measure.	Translation, rotation, or reflection.	A dilation with scale factor 1/2.
Line symmetry	A reflection carries a figure onto itself.	Fold mentally across a candidate line and match every point.	A line that splits area equally but does not match corresponding points.
Rotational symmetry	A rotation less than 360° carries a figure onto itself.	Rotate about the center and check whether every point lands on the figure.	A turn that changes which outline is occupied.
Order of rotational symmetry	The number of matching positions in one full turn, including the starting position.	For a regular n-gon, the order is n.	The number of nontrivial rotation angles listed.

Worked examples

Example 1 — the full symmetry set of a regular hexagon

Describe every nontrivial rotation and every reflection that carries a regular hexagon onto itself.

Suggested work

Givens

regular hexagon 6 equal central sectors

Tools

smallest rotational step = 360° × 1/6

Primitives

center of the hexagon opposite vertex pairs opposite side pairs

Computation

360° × 1/6 = 60° Nontrivial rotations: 60°, 120°, 180°, 240°, 300° Reflection lines: 3 lines through pairs of opposite vertices and 3 lines through midpoints of pairs of opposite sides. The hexagon has 6 reflection lines and rotational symmetry of order 6.

Notice: The rotational order includes the starting position, while the nontrivial angle list does not include 0°.

Example 2 — a non-square rectangle

Describe the rotations and reflections that carry a non-square rectangle onto itself.

Suggested work

Givens

rectangle with unequal side lengths

Tools

Test rotations about the center. Test lines through the center.

Primitives

horizontal side length ≠ vertical side length

Computation

A 180° rotation pairs each vertex with the opposite vertex, so the outline matches. A 90° rotation exchanges the unequal side lengths, so the outline does not match. A vertical center line reflects left to right. A horizontal center line reflects top to bottom. Nontrivial rotations: 180° Reflection lines: 2 Rotational order: 2

Notice: The unequal side lengths rule out quarter-turn symmetry.

Example 3 — an isosceles triangle that is not equilateral

Describe the rotations and reflections that carry an isosceles triangle that is not equilateral onto itself.

Suggested work

Givens

two equal sides third side has a different length

Tools

Test the line through the apex and midpoint of the base. Test rotations about the center.

Primitives

apex base midpoint

Computation

Reflection across the line through the apex and base midpoint swaps the equal sides and fixes the apex and base midpoint. No rotation less than 360° matches all three vertices because the base and apex roles are different. Nontrivial rotations: none Reflection lines: 1 Rotational order: 1

Common traps

A side does not guarantee a symmetry line

Counting sides does not determine reflection symmetry for every figure. A non-square rectangle has four sides but only two reflection lines. Test each candidate line by matching every point.

Practice

List the nontrivial rotation angles and reflection lines of a square.
Answer — tap to show
Rotations: 90°, 180°, and 270°. Reflection lines: 4, consisting of the 2 diagonals and the 2 lines through midpoints of opposite sides. Rotational order: 4.
Suggested work — tap to show
Givens
square 4 equal central sectors
Tools
smallest turn = 360° × 1/4
Primitives
diagonals midpoints of opposite sides
Computation
360° × 1/4 = 90° Multiples less than 360°: 90°, 180°, 270° Two diagonal reflection lines + two midpoint reflection lines = 4 reflection lines
State the rotational order, nontrivial rotation angles, and number of reflection lines for a regular pentagon.
Answer — tap to show
Rotational order: 5. Nontrivial rotations: 72°, 144°, 216°, and 288°. Reflection lines: 5.
A classmate writes, A rectangle has four sides, so every rectangle has four lines of symmetry. Diagnose the error and give the correct line count for a non-square rectangle.
Answer — tap to show
The classmate used side count as line count without testing reflections. A non-square rectangle has 2 reflection lines: the horizontal and vertical lines through its center.
Suggested work — tap to show
Givens
claim: 4 sides → 4 reflection lines
Tools
A reflection line must map every point of the figure onto the figure.
Primitives
test center lines test diagonals
Computation
The horizontal and vertical center lines match the rectangle. A diagonal reflection exchanges the unequal length and width, so the image does not match. Correct count: 2.
Describe all symmetries of an equilateral triangle.
Answer — tap to show
Nontrivial rotations: 120° and 240°. Reflection lines: 3, each through one vertex and the midpoint of the opposite side. Rotational order: 3.
Suggested work — tap to show
Givens
equilateral triangle 3 equal central sectors
Tools
smallest turn = 360° × 1/3
Primitives
each vertex midpoint of opposite side
Computation
360° × 1/3 = 120° Nontrivial turns: 120°, 240° One reflection line per vertex gives 3 reflection lines.
A rhombus is not a square. Describe its rotational and reflection symmetries.
Answer — tap to show
It has one nontrivial rotation, 180°, and two reflection lines, its diagonals. Its rotational order is 2.
Suggested work — tap to show
Givens
rhombus, not a square
Tools
Test 180° about the center. Test each diagonal as a reflection line.
Primitives
opposite vertices diagonals
Computation
A 180° rotation swaps opposite vertices and matches the outline. Reflection across either diagonal swaps the two vertices not on that diagonal and fixes the diagonal's endpoints. A 90° rotation does not preserve a non-square rhombus.
Describe every nontrivial rotation and classify every reflection line of a regular octagon.
Answer — tap to show
Nontrivial rotations: 45°, 90°, 135°, 180°, 225°, 270°, and 315°. It has 8 reflection lines: 4 through pairs of opposite vertices and 4 through midpoints of pairs of opposite sides. Rotational order: 8.
Suggested work — tap to show
Givens
regular octagon 8 equal central sectors
Tools
smallest turn = 360° × 1/8
Primitives
opposite vertex pairs opposite side pairs
Computation
360° × 1/8 = 45° List the seven positive multiples of 45° less than 360°. Four opposite-vertex axes + four opposite-side-midpoint axes = 8 reflection lines.