Symmetry is a property of shapes where one part is a mirror image of the other.
Key Concepts:
- Line Symmetry: A shape has line symmetry if it can be folded along a line (line of symmetry) so that the two halves match exactly.
- Rotational Symmetry: A shape has rotational symmetry if it looks exactly the same after a rotation of less than 360 degrees about a fixed point (center of rotation).
- Order of Rotational Symmetry: The number of times a shape coincides with itself in one complete turn (360 degrees).
Examples of Symmetrical Shapes:
- Equilateral triangle: 3 lines of symmetry, order of rotational symmetry is 3.
- Square: 4 lines of symmetry, order of rotational symmetry is 4.
- Circle: Infinite lines of symmetry, infinite order of rotational symmetry.
Real-world Applications:
Symmetry is found in nature, art, architecture, and many other areas. It’s a concept that helps us understand patterns and beauty in the world around us.
Exercise 12.1
1. Copy the figures with punched holes and find the axis of symmetry for the following:
Ans :
2. Give the line(s) of symmetry, find the other hole(s):
Ans :
3. In the following figures, the mirror line (i.e., the line of symmetry) is given as dotted line. Complete each figure performing reflection in the dotted (mirror) line. (You might perhaps place a mirror along the dotted line and look into the mirror for the image). Are you able to recall the name of the figure you complete?
Ans :
Identifying the Figures:
- (a) Rectangle
- (b) Triangle
- (c) Triangle
- (d) Circle
- (e) Pentagon
- (f) Hexagon
4. The following figures have more than one line of symmetry. Such figures are said to have multiple lines of symmetry.
Identify multiple lines of symmetry, if any, in each of the following figures:
Ans :
Here, figure (a), (d), (e), (g) and (h) are the multiple lines of symmetry.
5. Copy the figure given here.
Take any one diagonal as a line of symmetry and shade a few more squares to make the figure symmetric about a diagonal. Is there more than one way to do that? Will the figure be symmetric about both the diagonals?
Ans :
6. Copy the diagram and complete each shape to be symmetric about the mirror line(s).
Ans :
7. State the number of lines of symmetry for the following figures:
(а) An equilateral triangle
(b) An isosceles triangle
(c) A scalene triangle
(d) A square
(e) A rectangle
(f) A rhombus
(g) A parallelogram
(h) A quadrilateral
(i) A regular hexagon
(j) A circle
Ans :
Number of Lines of Symmetry:
- (a) An equilateral triangle: 3
- (b) An isosceles triangle: 1
- (c) A scalene triangle: 0
- (d) A square: 4
- (e) A rectangle: 2
- (f) A rhombus: 2
- (g) A parallelogram: 0
- (h) A quadrilateral: Generally 0, but can have more depending on its shape (e.g., rectangle, square)
- (i) A regular hexagon: 6
- (j) A circle: Infinite
8. What letters of the English alphabet have reflectional symmetry (i.e. symmetry related to mirror reflection) about
(а) a vertical mirror
(b) a horizontal mirror
(c) both horizontal and vertical mirrors.
Ans :
(a) Vertical mirror:
- Letters with vertical symmetry: A, H, I, M, O, T, U, V, W, X, Y
(b) Horizontal mirror:
- Letters with horizontal symmetry: B, C, D, E, H, I, K, O, X
(c) Both horizontal and vertical mirrors:
- Letters with both symmetries: H, I, O, X
9. Give three examples of shapes with no line of symmetry.
Ans :
- Scalene Triangle: A triangle with all sides of different lengths has no line of symmetry.
- Parallelogram: A parallelogram, unless it’s a special case like a rectangle or rhombus, does not have a line of symmetry.
- Trapezium: A trapezium, which is a quadrilateral with only one pair of parallel sides, generally does not have a line of symmetry.
10. What other name can you give of the line of symmetry of
(a) an isosceles triangle?
(b) a circle?
Ans :
(a) Isosceles triangle
- Median: The line of symmetry in an isosceles triangle is also the median to the base. A median is a line segment that joins a vertex to the midpoint of the opposite side.
- Altitude: In an isosceles triangle, the line of symmetry is also the altitude to the base. An altitude is a perpendicular line segment from a vertex to the opposite side.
(b) Circle
- Diameter: Any line passing through the center of a circle is a line of symmetry and is also a diameter. A diameter is a line segment that joins two points on the circle and passes through the center.
Exercise 12.2
1. Which of the following figures have rotational symmetry of order more than 1?
Ans :
The figures with rotational symmetry of order more than 1 are (a), (b), (d), (e), and (f).
2. Give the order of rotational symmetry for each figure:
Ans :
- (a): 2
- (b): 2
- (c): 3
- (d): 4
- (e): 4
- (f): 5
- (g): 6
- (h): 1
Exercise 12.3
1. Name any two figures that have both line symmetry and rotational symmetry.
Ans :
Square: A square has four lines of symmetry and rotational symmetry of order 4.
Equilateral triangle: An equilateral triangle has three lines of symmetry and rotational symmetry of order 3
2. Draw, wherever possible, a rough sketch of
(i) a triangle with both line and rotational symmetries of order more than 1.
(ii) a triangle with only line symmetry and no rotational symmetry of order more than 1.
(iii) a quadrilateral with a rotational symmetry of order more than 1 but not a line symmetry.
(iv) a quadrilateral with line symmetry but not a rotational symmetry of order more than 1.
Ans :
(i)
(ii) Not Possible
(iii)
(iv) Not Possible
3. If a figure has two or more lines of symmetry, should it have rotational symmetry of order more than 1?
Ans :
Yes, if a figure has two or more lines of symmetry, it must have rotational symmetry of order more than 1.
4. Fill in the blanks:
| Shape | Centre of rotation | Order of rotation | Angle of rotation |
| Square | |||
| Rectangle | |||
| Rhombus | |||
| Equilateral triangle | |||
| Regular hexagon | |||
| Circle | |||
| Semicircle |
Ans :
| Shape | Centre of Rotation | Order of Rotation | Angle of Rotation |
| Square | Centre of the square | 4 | 90° |
| Rectangle | Centre of the rectangle | 2 | 180° |
| Rhombus | Centre of the rhombus | 2 | 180° |
| Equilateral triangle | Centre of the triangle | 3 | 120° |
| Regular hexagon | Centre of the hexagon | 6 | 60° |
| Circle | Centre of the circle | Infinite | Any angle |
| Semicircle | No rotational symmetry | – | – |
5. Name the quadrilaterals which have both line and rotational symmetry of order more than 1.
Ans :
The quadrilaterals that possess both line symmetry (more than one) and rotational symmetry of order more than 1 are:
- Square
- Rectangle
- Rhombus
6. After rotating by 60° about a centre, a figure looks exactly the same as its original position. At what other angles will this happen for the figure?
Ans :
If a figure looks exactly the same after rotating by 60° about a center, it will also look the same at angles that are multiples of 60°.
This is because the figure is essentially returning to its original position after each 60° rotation.
So, the other angles at which the figure will look the same are:
- 120°
- 180°
- 240°
- 300°
- 360°
7. Can we have a rotational symmetry of order more than 1 whose angle of rotation is
(i) 45°?
(ii) 17°?
Ans :
(i)Yes, we can have a rotational symmetry of order more than 1 with a rotation angle of 45°.
(ii) No, we cannot have a rotational symmetry of order more than 1 with a rotation angle of 17°.
