Question

1. select all the angles of rotation that produce symmetry for this flower.

a. 45
b. 60
c. 90
d. 120
e. 135
f. 150
g. 180
(from unit 1, lesson 16.)

1. three line segments form the letter n. rotate the letter n clockwise around the midpoint of segment bc by 180 degrees. describe the result.

Explanation:

Step1: Determine the number of petals

Assume the flower has 8 - fold symmetry (by visual inspection of the rotational symmetry pattern). The formula for the angle of rotational symmetry is \(\theta=\frac{360^{\circ}}{n}\), where \(n\) is the number of times the figure coincides with itself in a full - rotation (\(360^{\circ}\)).
If \(n = 8\), then \(\theta=\frac{360^{\circ}}{8}=45^{\circ}\). Multiples of \(45^{\circ}\) that are less than or equal to \(360^{\circ}\) will also give rotational symmetry.
For \(n = 4\) (another way to check, since \(45\times4 = 180\)), \(\theta=\frac{360^{\circ}}{4}=90^{\circ}\), and \(\frac{360^{\circ}}{2}=180^{\circ}\)

Step2: Check each option

• Option A: \(\frac{360}{45}=8\). A rotation of \(45^{\circ}\) will map the flower onto itself.
• Option B: \(\frac{360}{60} = 6\). Since the flower does not have 6 - fold symmetry (from visual inspection of the given figure), a \(60^{\circ}\) rotation will not map the flower onto itself.
• Option C: \(\frac{360}{90}=4\). A rotation of \(90^{\circ}\) will map the flower onto itself.
• Option D: \(\frac{360}{120}=3\). Since the flower does not have 3 - fold symmetry (from visual inspection of the given figure), a \(120^{\circ}\) rotation will not map the flower onto itself.
• Option E: \(\frac{360}{135}=\frac{8}{3}\). A non - integer result, so a \(135^{\circ}\) rotation will not map the flower onto itself.
• Option F: \(\frac{360}{150}=\frac{12}{5}\). A non - integer result, so a \(150^{\circ}\) rotation will not map the flower onto itself.
• Option G: \(\frac{360}{180}=2\). A rotation of \(180^{\circ}\) will map the flower onto itself.

Answer:

A. \(45\), C. \(90\), G. \(180\)