Symmetry

A figure has line symmetry if it can be folded along a line so that the two parts match exactly, like the two halves of a butterfly. This line is the line or axis of symmetry, and each half is the mirror image of the other. A regular polygon, with equal sides and equal angles, has as many lines of symmetry as it has sides: an equilateral triangle has 3, a square 4 and a regular hexagon 6. A scalene triangle has no line of symmetry, while a circle has infinitely many. The chapter also introduces rotational symmetry. A figure has rotational symmetry if it looks exactly the same after turning through less than a full turn about a fixed point, the centre of rotation. The number of times it fits onto itself in one complete turn is its order, and the smallest angle of turn is 360 degrees divided by the order. A square has order 4 with angle 90 degrees, while windmill blades and letters such as S, N and Z also have rotational symmetry.

Detailed Worksheet: Symmetry Section A - Definitions (10 marks) 1. What is line symmetry? Give two examples from nature. (2 marks) 2. What is a regular polygon? How many lines of symmetry does a regular polygon with n sides have? (2 marks) 3. Define rotational symmetry and the centre of rotation. (2 marks) 4. What is meant by the order of rotational symmetry and the angle of rotation? (2 marks) 5. Every figure has rotational symmetry of order 1. Explain what this means. (2 marks) Section B - Calculations and Applications (15 marks) 6. Write the number of lines of symmetry of (i) an equilateral triangle (ii) a square (iii) a rectangle (iv) a rhombus (v) a regular pentagon (vi) a regular hexagon. (3 marks) 7. Find the angle of rotational symmetry for figures of order (i) 2 (ii) 3 (iii) 4 (iv) 5 (v) 6 (vi) 8. (3 marks) 8. A figure has a smallest angle of rotation of 60 degrees. (i) Find its order. (ii) Name a shape with this order. (iii) Through what other angles less than 360 degrees will it look the same? (3 marks) 9. Write the order of rotational symmetry of (i) a circle (ii) an equilateral triangle (iii) a parallelogram (iv) a rectangle (v) a regular octagon (vi) a scalene triangle. (3 marks) 10. Which capital letters among A, H, I, N, O, S, X, Z have (i) only line symmetry (ii) only rotational symmetry of order more than 1 (iii) both? (3 marks) Section C - Diagrams (10 marks) 11. Draw (i) an equilateral triangle (ii) a square (iii) a regular hexagon (iv) a rectangle and mark all their lines of symmetry with dotted lines. (4 marks) 12. Draw a figure with exactly one line of symmetry and a figure with exactly two lines of symmetry. Mark the lines. (3 marks) 13. Draw a four-blade windmill (or fan) figure. Mark its centre of rotation and show the positions after quarter turns. State its order. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. (i) Can a figure have rotational symmetry of order more than 1 but no line of symmetry? Give an example. (ii) Can a figure have line symmetry but no rotational symmetry of order more than 1? Give an example. (iii) Name two figures with both. (iv) What is special about a circle? (5 marks) 15. Complete the figure: half of a symmetrical shape is drawn on squared paper on one side of a dotted line. (i) Explain how you would complete it. (ii) What is the rule about distances from the mirror line? (iii) If a point is 3 squares to the left of the line, where is its image? (iv) Why does the completed figure have line symmetry? (5 marks) 16. A rangoli design looks the same after turning through 45 degrees about its centre. (i) Find its order of rotational symmetry. (ii) List all the angles less than 360 degrees at which it looks the same. (iii) If it also has line symmetry, how many lines can it have if it is based on a regular octagon? (iv) Why are such designs pleasing? (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Use dotted lines for axes of symmetry and mark the centre of rotation with a dot.