Question

3 select all the angles of rotation that produce symmetry for this flower. a 45° b 90° c 135° d 180° e 225° f 270°

Explanation:

Step1: Recall rotation - symmetry concept

A shape has rotation - symmetry if it looks the same after a rotation of an angle $\theta$ where $0<\theta\leq360^{\circ}$. For a regular $n$ - sided figure, the angles of rotation that produce symmetry are given by $\theta=\frac{360^{\circ}}{n}k$, where $k = 1,2,\cdots,n - 1$. If the flower has 8 equal - parts (assuming a regular 8 - petal flower), then the angles of rotation that produce symmetry are $\frac{360^{\circ}}{8}=45^{\circ}$, and its multiples less than or equal to $360^{\circ}$.

Step2: Check each option

• Option A: $45^{\circ}=\frac{360^{\circ}}{8}\times1$, so a $45^{\circ}$ rotation will produce symmetry.
• Option B: $90^{\circ}=\frac{360^{\circ}}{8}\times2$, so a $90^{\circ}$ rotation will produce symmetry.
• Option C: $135^{\circ}=\frac{360^{\circ}}{8}\times3$, so a $135^{\circ}$ rotation will produce symmetry.
• Option D: $180^{\circ}=\frac{360^{\circ}}{8}\times4$, so a $180^{\circ}$ rotation will produce symmetry.
• Option E: $225^{\circ}=\frac{360^{\circ}}{8}\times5$, so a $225^{\circ}$ rotation will produce symmetry.
• Option F: $270^{\circ}=\frac{360^{\circ}}{8}\times6$, so a $270^{\circ}$ rotation will produce symmetry.

Answer:

A. $45^{\circ}$, B. $90^{\circ}$, C. $135^{\circ}$, D. $180^{\circ}$, E. $225^{\circ}$, F. $270^{\circ}$