Question

for each figure below, determine the order of rotational symmetry and the angles of rotation required to map the image onto itself. 6. 7. 8. 9. 10. 11. 12. a regular polygon has an angle of rotational symmetry of 18°. how many sides does the polygon have

Explanation:

Step1: Recall the formula for order of rotational symmetry

For a regular polygon, the order of rotational symmetry is equal to the number of sides.

Step2: Recall the formula for angle of rotation

The angle of rotation $\alpha$ for a regular polygon with $n$ sides is given by $\alpha=\frac{360^{\circ}}{n}$, and the non - zero angles of rotation that map the polygon onto itself are $k\alpha$ where $k = 1,2,\cdots,n - 1$.

Step3: Analyze rectangle (6)

A rectangle has 2 lines of rotational symmetry. The non - zero angle of rotation that maps it onto itself is $180^{\circ}$.

Step4: Analyze circle (7)

A circle can be rotated by any non - negative angle and still map onto itself, so it has infinite order of rotational symmetry.

Step5: Analyze equilateral triangle (8)

It has 3 sides, so order of rotational symmetry is 3. $\alpha=\frac{360^{\circ}}{3}=120^{\circ}$, and angles are $120^{\circ},240^{\circ}$.

Step6: Analyze 5 - petal flower (9)

It has 5 sides (petals), so order is 5. $\alpha=\frac{360^{\circ}}{5}=72^{\circ}$, and angles are $72^{\circ},144^{\circ},216^{\circ},288^{\circ}$.

Step7: Analyze regular octagon (10)

It has 8 sides, so order is 8. $\alpha=\frac{360^{\circ}}{8}=45^{\circ}$, and angles are $45^{\circ},90^{\circ},135^{\circ},180^{\circ},225^{\circ},270^{\circ},315^{\circ}$.

Step8: Analyze square (11)

It has 4 sides, so order is 4. $\alpha=\frac{360^{\circ}}{4}=90^{\circ}$, and angles are $90^{\circ},180^{\circ},270^{\circ}$.

Step9: Solve for number of sides (12)

Given $\alpha = 18^{\circ}$, using $n=\frac{360^{\circ}}{\alpha}$, we get $n=\frac{360^{\circ}}{18^{\circ}}=20$.

Answer:

1. Order of rotational symmetry: 2; Angles of rotation: $180^{\circ}$
2. Order of rotational symmetry: infinite; Angles of rotation: Any non - negative angle $\theta$ where $0^{\circ}<\theta < 360^{\circ}$
3. Order of rotational symmetry: 3; Angles of rotation: $120^{\circ},240^{\circ}$
4. Order of rotational symmetry: 5; Angles of rotation: $72^{\circ},144^{\circ},216^{\circ},288^{\circ}$
5. Order of rotational symmetry: 8; Angles of rotation: $45^{\circ},90^{\circ},135^{\circ},180^{\circ},225^{\circ},270^{\circ},315^{\circ}$
6. Order of rotational symmetry: 4; Angles of rotation: $90^{\circ},180^{\circ},270^{\circ}$
7. 20