What is line (axial) symmetry?
Fold a figure along a line. If the two halves lie exactly on top of each other, the figure has line symmetry. The fold line is called the axis or line of symmetry.
Each point has a mirror image point. Both points are the same distance from the axis. The segment joining them crosses the axis at a right angle (90°).
- On squared paper: if the axis is vertical, a point 3 squares to the left goes 3 squares to the right. It stays on the same row.
- With coordinates: reflecting in the y-axis changes (x, y) to (−x, y). Reflecting in the x-axis changes (x, y) to (x, −y).
In nature and design you see it everywhere: leaves, faces, butterflies, kites, temple gates, logos.
Perpendicular bisector and angle bisector
Perpendicular bisector
The perpendicular bisector of segment AB is the line through its midpoint that makes 90° with AB. It is the line of symmetry of the segment.
Property: every point on it is the same distance from A and from B. And any point that is equally far from A and B lies on it.
Angle bisector
The angle bisector splits an angle into two equal angles. It is the line of symmetry of the angle.
Property: every point on it is the same distance from the two arms (distance is measured along a perpendicular). Points inside the angle that are equally far from both arms lie on it.
These ideas solve many problems: finding a point equally far from two villages, the centre of a circle through three points, or the centre of a circle touching all sides of a triangle.
How many lines of symmetry? Completing a figure
| Figure | Lines of symmetry |
|---|---|
| Scalene triangle | 0 |
| Isosceles triangle | 1 |
| Equilateral triangle | 3 |
| Rectangle (not square) | 2 |
| Rhombus (not square) | 2 (the diagonals) |
| Square | 4 |
| Regular polygon with n sides | n |
| Circle | infinitely many |
To complete a figure given half of it and the axis: take each corner, measure its distance to the axis at 90°, mark the same distance on the other side, then join the new points in the same order.
Point (central) symmetry
A figure has point symmetry (central symmetry) if turning it 180° about a point O gives the same figure. O is the centre of symmetry.
The image of point P is P′ where O is the midpoint of PP′. So OP = OP′ and P, O, P′ are on one line. With O at the origin, (x, y) becomes (−x, −y).
- Have a centre: parallelogram (where diagonals meet), rectangle, rhombus, square, circle, regular hexagon, letters S, N, Z.
- No centre: triangle, kite, trapezium (usually), letter A.
Median doubling: in triangle ABC with median AM, extend AM past M to D so that MD = AM. Then D is the image of A under central symmetry about M, and ABDC is a parallelogram. This trick helps prove facts about medians.
Try it: ink-blot and turn test
Fold a sheet of paper, put a few drops of ink or paint near the fold, press and open it: you made a line-symmetric picture. Now write your name in capitals and test each letter: does it fold onto itself (line), or look the same turned upside down (point)? Then check your guesses in the 3D free play.
Key formulas and definitions
- Reflection in the y-axis: (x, y) → (−x, y)
- Reflection in the x-axis: (x, y) → (x, −y)
- Reflection in the line y = x: (x, y) → (y, x)
- Point symmetry about the origin: (x, y) → (−x, −y)
- Point symmetry about O(a, b): (x, y) → (2a − x, 2b − y)
- Regular n-gon: n lines of symmetry
Worked examples
1. Reflect the point (3, 5) in the y-axis and in the x-axis.
In the y-axis only x changes sign: (−3, 5). In the x-axis only y changes sign: (3, −5).
2. How many lines of symmetry does a regular hexagon have? Does it have a centre of symmetry?
A regular polygon with n sides has n lines: 6 lines. Turning it 180° gives the same hexagon, so yes, it has a centre (its middle).
3. A(1, 2) and B(7, 2). Find the perpendicular bisector of AB and check that P(4, 6) is equally far from A and B.
Midpoint M = (4, 2). AB is horizontal, so the bisector is the vertical line x = 4. P is on it. PA = √(3² + 4²) = 5, PB = √(3² + 4²) = 5. Equal.
4. Find the image of P(2, −3) under point symmetry about O(1, 1).
P′ = (2·1 − 2, 2·1 − (−3)) = (0, 5). Check: midpoint of PP′ = ((2+0)/2, (−3+5)/2) = (1, 1) = O.
5. A point is inside angle XOY = 70° and is 4 cm from OX and 4 cm from OY. What angle does OP make with OX?
Equal distances from both arms means P is on the bisector. So angle POX = 70° ÷ 2 = 35°.
6. Two villages A and B lie 6 km apart. A well must be equally far from both and 4 km from the road AB. How far is it from each village?
The well is on the perpendicular bisector, 4 km from the midpoint. Distance = √(3² + 4²) = 5 km from each village.
Common mistakes
- Calling a diagonal of a rectangle a line of symmetry. Folding along it does not match the halves.
- Measuring the distance to the axis slantwise instead of at 90°.
- Mixing up point symmetry with reflection: point symmetry changes BOTH coordinates, (x, y) → (−x, −y).
- Thinking every parallelogram has lines of symmetry. It has a centre but usually no line of symmetry.