Solutions: Page 235

Questions (Page 235)

Can you draw a figure with radial arms that has:

• a) Exactly 5 angles of symmetry:

  • Draw a figure with 5 equally spaced arms (like a starfish).
  • Angle: $360^{\circ} \div 5 = 72^{\circ}$.
  • Angles of Symmetry: $72^{\circ}, 144^{\circ}, 216^{\circ}, 288^{\circ}, 360^{\circ}$.

• b) 6 angles of symmetry:

  • Draw a figure with 6 equally spaced arms (like a snowflake).
  • Angle: $360^{\circ} \div 6 = 60^{\circ}$.
  • Angles of Symmetry: $60^{\circ}, 120^{\circ}, 180^{\circ}, 240^{\circ}, 300^{\circ}, 360^{\circ}$.

Consider a figure with radial arms having exactly 7 angles of symmetry.

• Smallest angle: $360^{\circ} \div 7$.
• Calculation: $360 \div 7 = 51$ with a remainder of 3.
• Answer: $51 \frac{3}{7}^{\circ}$. It is not a whole number.

Figure it Out

1. Find the angles of symmetry for the given figures:

• (a) Plus Sign (+): Order 4.
  • Angles: $90^{\circ}, 180^{\circ}, 270^{\circ}, 360^{\circ}$.
• (b) H-shape: Order 2.
  • Angles: $180^{\circ}, 360^{\circ}$ (Assuming simple H rotation). Note: If the text refers to a specific radial figure, check if it’s 1-fold (only 360). The solution key says “Angle of symmetry = 360” for figure (b), implying it has NO rotational symmetry other than the full turn.
• (c) Line segment or I-shape: Order 2.
  • Angles: $180^{\circ}, 360^{\circ}$.

2. Which figures have more than one angle of symmetry?

• Circle with Cross: Yes (Order 4).
• Equilateral Triangle: Yes (Order 3).
• Circle with 3 sectors: Yes (Order 3).
• Fan (4 blades): Yes (Order 4).
• X shape: Yes (Order 2 or 4 depending on angles).
• Star (5 points): Yes (Order 5).