Solutions: Page 238

Figure it Out (Page 238)

1. Colour the sectors of the circle: The circle is divided into 12 sectors.

• i) 3 angles of symmetry: You need Order 3. $12 \div 3 = 4$. You must repeat a pattern every 4 sectors.
  • Example: Color sectors 1, 5, 9.
• ii) 4 angles of symmetry: You need Order 4. $12 \div 4 = 3$. Repeat pattern every 3 sectors.
  • Example: Color sectors 1, 4, 7, 10.
• iii) Possible numbers of angles:
  • The factors of 12 are 1, 2, 3, 4, 6, 12. You can create symmetries of these orders.

2. Draw figures other than a circle and square that have both reflection and rotational symmetry.

• Regular Pentagon: 5 lines of symmetry, Order 5 rotation.
• Regular Hexagon: 6 lines of symmetry, Order 6 rotation.
• Rectangle: 2 lines of symmetry, Order 2 rotation ($180^{\circ}$).

3. Draw rough sketches:

• a. Triangle with at least 2 lines and 2 angles: Equilateral Triangle (3 lines, 3 angles).
• b. Triangle with 1 line but no rotational symmetry: Isosceles Triangle.
• c. Quadrilateral with rotational symmetry but no reflection: Parallelogram.
• d. Quadrilateral with reflection but no rotational: Isosceles Trapezoid or a Kite.

4. Smallest angle is 60°. What are the others? Multiples of 60: $120^{\circ}, 180^{\circ}, 240^{\circ}, 300^{\circ}, 360^{\circ}$.

5. 60° is an angle. Two angles are less than 60°. What is the smallest? If 60 is a symmetry angle, the smallest angle must be a factor of 60. If there are exactly two angles less than 60 (say $x$ and $2x$), and $3x = 60$, then $x=20$. Angles: $20^{\circ}, 40^{\circ}, 60^{\circ} \dots$ Answer: $20^{\circ}$.

6. Can a figure have rotational symmetry with smallest angle:

• a. 45°? Yes, because $360 \div 45 = 8$ (whole number). It would be an octagon.
• b. 17°? No, because $360 \div 17 = 21.17$ (not a whole number).