Symmetry
Explore symmetry, line of symmetry, mirror halves, reflection symmetry, rotational symmetry, angles of symmetry, and symmetries of a circle.
9.1 Line of Symmetry
When a figure is made up of parts that repeat in a definite pattern, we say that the figure has symmetry. A line that cuts a plane figure into two parts that exactly overlap when folded along that line is called a line of symmetry or axis of symmetry of the figure. These overlapping halves are called mirror halves.
Figure 9.1: Lines of Symmetry in a Square
A square has 4 lines of symmetry: 1 vertical, 1 horizontal, and 2 diagonals.
Generating Symmetrical Shapes
- Ink Blot Devils: Spill ink drops on one half of a folded paper, press together, and open to get a symmetrical pattern along the central fold line.
- Paper Folding & Cutting: Fold paper once or multiple times, make cuts, and unfold to create symmetrical designs.
- Punching Game: Punch holes in a folded sheet; when unfolded, holes appear symmetrically across the fold lines.
9.2 Rotational Symmetry
When a figure looks exactly the same when rotated by an angle about a fixed point, it is said to have rotational symmetry. The fixed point is called the centre of rotation, and the angle is called an angle of rotational symmetry.
Interactive Rotational Symmetry Simulator
Rotate the square below to observe how it matches its original shape at every $90^\circ$ turn:
Original Position ($0^\circ$)
Symmetries of a Circle
- Infinite Reflection Lines: Every diameter of a circle is a line of reflection symmetry.
- Infinite Angles of Rotational Symmetry: Rotating a circle clockwise or counter-clockwise about its centre by any angle coincides with itself.
Chapter Summary
- A figure is symmetrical if it is composed of parts repeating in a definite pattern.
- A line of symmetry divides a figure into two mirror halves that overlap completely when folded.
- A figure can have no lines of symmetry, exactly 1 line, or multiple lines of symmetry.
- An angle of symmetry is an angle strictly between $0^\circ$ and $360^\circ$ by which a figure rotates about its centre to look identical to its initial state.
- Smallest angle of symmetry for regular shapes divides $360^\circ$ evenly (e.g., $360^\circ / n$).
- A circle has infinitely many lines of symmetry (diameters) and infinitely many angles of rotational symmetry.
Chapter Solutions
Q1. Do you see any line of symmetry in the figures at the start of the chapter? What about the cloud?
Ans: Yes, there are 6, 4, and 1 lines of symmetry in the figures of the flower, rangoli, and butterfly respectively. There is no line of symmetry in the figures of the pinwheel and cloud.
Q. How many lines of symmetry does a square shape have? Is a rectangle's diagonal a line of symmetry?
Ans: A square shape has 4 lines of symmetry (vertical, horizontal, and 2 diagonals). No, a rectangle's diagonal is not a line of symmetry.
Q. How many lines of symmetry do these shapes have?
(a) Square / Octagon pattern: 4 and 8 lines of symmetry.
(b) Equilateral Triangle (equal sides & angles): 3 lines of symmetry.
(c) Regular Hexagon (equal sides & angles): 6 lines of symmetry.
Q. Draw/describe triangles with (a) 1 line of symmetry, (b) 3 lines of symmetry, (c) 0 lines of symmetry. Is it possible to draw a triangle with exactly 2 lines of symmetry?
Ans:
(a) Isosceles Triangle has exactly 1 line of symmetry.
(b) Equilateral Triangle has exactly 3 lines of symmetry.
(c) Scalene Triangle has 0 lines of symmetry.
No, it is not possible to draw a triangle with exactly 2 lines of symmetry.
Q. What are the angles of symmetry for radial arm figures with (a) 5 arms, (b) 6 arms?
Ans:
(a) 5 Radial Arms: Angle between adjacent arms $= 360^\circ / 5 = 72^\circ$.
Angles of symmetry $= 72^\circ, 144^\circ, 216^\circ, 288^\circ, 360^\circ$.
(b) 6 Radial Arms: Angle between adjacent arms $= 360^\circ / 6 = 60^\circ$.
Angles of symmetry $= 60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ, 360^\circ$.
Q4. In a figure, $60^\circ$ is the smallest angle of symmetry. What are the other angles of symmetry?
Ans: The other angles are multiples of $60^\circ$: $120^\circ, 180^\circ, 240^\circ, 300^\circ, 360^\circ$.
Q6. Can a figure have a smallest angle of rotational symmetry of (a) $45^\circ$, (b) $17^\circ$?
Ans:
(a) Yes, because $360^\circ / 45^\circ = 8$ (a whole number).
(b) No, because $360^\circ / 17^\circ = 21.176...$ (not a natural number).