Symmetry

1Summary 重點摘要

Reflectional symmetry is like a mirror: fold along the line and both halves match. Rotational symmetry counts how often a shape matches itself in one full turn.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. What is symmetry?Symmetry is the property that a figure can be folded along a line so that the two halves match exactly. The most familiar example is a butterfly, whose left and right wings mirror each other pattern for pattern; a human face is roughly symmetric, and a heart-shaped card folds exactly along its centre line. Symmetry is widely used in design — school badges, stamps and building facades gain stability from it. Judging symmetry needs the fold test or counting axes, not impression: the letter R looks regular but has no line of symmetry at all.
2. 2. Line of symmetryA line of symmetry is the line that divides a figure into two halves which coincide exactly when folded, usually drawn as a dashed line. The line down a butterfly's body and the centre line of a heart are examples. A figure may have several lines: a square has 4 and a circle infinitely many; it may have none, as with a scalene triangle. When searching, try every direction — horizontal, vertical, and along the diagonals — and record each axis found; stopping after one direction and concluding is the common mistake.
3. 3. How to check symmetry?To test whether a figure is symmetric, the fold test is the most reliable: first, guess and draw a possible line of symmetry; second, fold the figure along it, in the mind or with real paper; third, check whether the two halves coincide completely, including shape, size and pattern direction. Coincide means symmetric; otherwise try another direction. Beware the mirror trap: letters p and q are left-right reversed, so a figure with p on one side and q on the other looks similar but is not symmetric, because the reversed letters cannot coincide after folding.
4. 4. Symmetric figure examplesMemorise the line counts of standard figures: a square has 4 (horizontal, vertical and two diagonals); a rectangle has 2 (only horizontal and vertical — the diagonals do not count); an equilateral triangle has 3; an isosceles triangle has 1; a circle has infinitely many. The classic examination trap is the rectangle: folding along a diagonal gives two triangles of equal area that do not coincide, so the diagonal is not an axis of symmetry. Verify by folding or cutting paper rather than memorising numbers blindly.
5. 5. Not symmetric examplesA scalene triangle has three unequal sides and 0 lines of symmetry, and a general irregular shape has none either. Among letters, R, F, G and J have no axis of symmetry while A, H, O and X do; such examples appear constantly in multiple-choice questions testing true understanding. Do not be misled by a figure that looks 'roughly symmetric': test direction by direction, since one failed fold disqualifies that direction, and only when every direction fails is the figure asymmetric.
6. 6. Reflection (mirror)Reflection symmetry works like a mirror image: the mirror line is the axis of symmetry, and the two sides are mirror images — left and right reversed, top and bottom unchanged. Raise your right hand before a mirror and the reflection raises the left; the mirror image of the letter b is d, and of p is q. To draw a reflection, counting grid squares is the safe method: each point on one side matches a point on the other at the same perpendicular distance from the axis, so a point 3 squares from the axis reflects to a point 3 squares away.
7. 7. Draw symmetric figureTo draw a symmetric figure: first draw the axis of symmetry as a dashed line; second draw the given half on one side; third mark the mirror image of every key point on the other side at equal distance from the axis, then join the points. If a vertex of the left half sits 4 squares from the axis, its partner on the right must also sit 4 squares away, at the same height. Fold the paper afterwards to check for coincidence — the quickest self-check; miscounting squares or forgetting to keep the height unchanged are the two usual errors.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Symmetry: both sides same
• Line of symmetry: fold line
• Check: fold and see overlap
• Square 4 lines, rectangle 2 lines
• Equilateral 3, isosceles 1, circle infinite