SYMMETRY
9.1 Line of Symmetry
A shape has a line of symmetry if a line can be drawn to divide the shape into two identical halves that are mirror images of each other. These halves are called mirror halves. When you fold the shape along this line, the two parts will overlap perfectly.
Reflection Symmetry
A figure that has one or more lines of symmetry is said to have reflection symmetry. A rectangle has two lines of symmetry, while a square has four. However, the diagonal of a rectangle (that is not a square) is not a line of symmetry, but the diagonals of a square are lines of symmetry.
Figure it Out (Pages 219-228)
1. Identify the line(s) of symmetry (Page 219)
For the figures on page 219, you can describe the lines of symmetry as follows:
- Figure 1: No lines of symmetry.
- Figure 2: One vertical line of symmetry.
- Figure 3: One horizontal line of symmetry.
- Figure 4: No lines of symmetry.
- Figure 5: One diagonal line of symmetry.
2. Given the line of symmetry, find the other hole(s). (Page 224)
You need to place a hole on the opposite side, at the same distance from the fold line, to create a mirror image.
- a. The second hole will be on the bottom left corner.
- b. The second hole will be in the top right section.
- c. Two more holes are needed: one on the right side of the vertical line and another reflecting the first hole across the horizontal line.
- d & e. The second hole will be on the opposite side of the circle, reflected across the line of symmetry.
3. How many lines of symmetry do these shapes have? (Page 226)
- a. (Square): 4 lines of symmetry (2 through the midpoints of opposite sides, 2 along the diagonals).
- a. (Star): 8 lines of symmetry (4 through opposite points, 4 through opposite valleys).
- b. (Equilateral Triangle): 3 lines of symmetry (from each corner to the midpoint of the opposite side).
4. Draw triangles with specific lines of symmetry (Page 228)
- a. Exactly one line of symmetry: This is an isosceles triangle.
- b. Exactly three lines of symmetry: This is an equilateral triangle.
- c. No line of symmetry: This is a scalene triangle.
- Is two lines of symmetry possible for a triangle? No. If a triangle has two lines of symmetry, it must also have a third, making it equilateral.
9.2 Rotational Symmetry
A figure has rotational symmetry if it looks the same after being rotated by a certain angle around a fixed point. This fixed point is the centre of rotation, and the angle is the angle of rotational symmetry.
Key Concepts
- Every figure has rotational symmetry of 360° because it returns to its original position after a full turn.
- If a figure has meaningful rotational symmetry, its smallest angle of rotation must be a factor of 360. For example, a square has a smallest angle of 90°, and 360 / 90 = 4.
- The order of rotational symmetry is the number of times a figure looks the same during a full 360° rotation. A square has an order of 4.
Figure it Out (Pages 235-239)
1. Give the order of rotational symmetry for each figure: (Page 236)
- Top row (left to right): Order 2, Order 2, Order 6.
- Bottom row (left to right): Order 4 (Swastika), Order 2, Order 5 (Pentagon).
2. Draw a rough sketch of: (Page 238)
- a. Triangle with at least two lines and two angles of symmetry: An equilateral triangle (it has 3 of each).
- b. Triangle with one line of symmetry but no rotational symmetry: An isosceles triangle.
- c. Quadrilateral with rotational symmetry but no reflection symmetry: A parallelogram (that isn’t a rhombus or rectangle).
- d. Quadrilateral with reflection symmetry but not rotational symmetry: A kite or an isosceles trapezoid.
3. Can we have a figure with a smallest rotational angle of: (Page 238)
- a. 45°? Yes, because 360 is divisible by 45 (360 / 45 = 8). An octagon is an example.
- b. 17°? No, because 360 is not divisible by 17.
4. Symmetry of Famous Structures (Page 239)
- New Parliament Building: The outer boundary is a hexagon. It has 6 lines of reflection symmetry and rotational symmetry of order 6 (angles: 60°, 120°, 180°, 240°, 300°, 360°).
- Ashoka Chakra: It has 24 lines of reflection symmetry (through each spoke and between each pair of spokes) and rotational symmetry of order 24 (smallest angle is 360/24 = 15°).
SUMMARY
- 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 is called a line of symmetry. A figure can have multiple lines of symmetry.
- A figure has rotational symmetry if it looks the same after being rotated by an angle between 0° and 360°. The point of rotation is the centre of rotation.
- A figure may have multiple angles of symmetry. Some figures have only line symmetry, some have only rotational symmetry, and some have both.