Question

for items 1 - 3, use the regular pentagon shown.

1. how many lines of symmetry does the pentagon have?

a 0
b 1
c 3
d 5

1. what is the least angle of rotation in degrees that maps the pentagon onto itself?
2. does the pentagon have the following types of symmetry?

for items 4 - 5, use the figure shown.

1. what reflection(s) map the figure onto itself?

a ( r_m ) only
b ( r_n ) only
c ( r_m ) and ( r_n )
d none

1. which rotation maps the figure onto itself?

a ( r_{(45^{circ},p)} )
b ( r_{(90^{circ},p)} )
c ( r_{(180^{circ},p)} )
d ( r_{(270^{circ},p)} )

Explanation:

1.

A regular pentagon has 5 lines of symmetry. Each line of symmetry passes through a vertex and the mid - point of the opposite side.

2.

The formula for the least angle of rotation of a regular \(n\) - sided polygon is \(\frac{360^{\circ}}{n}\). For a pentagon \(n = 5\), so \(\frac{360^{\circ}}{5}=72^{\circ}\).

3.

• Reflectional symmetry: A regular pentagon has reflectional symmetry. The number of lines of reflectional symmetry is equal to the number of sides (\(n = 5\)). So, for reflectional symmetry, the answer is "Yes".
• Point symmetry: A polygon has point symmetry if it looks the same when rotated \(180^{\circ}\) about its center. A regular pentagon does not have point symmetry. If we rotate a regular pentagon \(180^{\circ}\) about its center, the vertices and sides do not map onto their original positions. So, for point symmetry, the answer is "No".

4.

The figure is symmetric about both line \(m\) (vertical line) and line \(n\) (horizontal line). A reflection over line \(m\) (\(r_m\)) and a reflection over line \(n\) (\(r_n\)) will map the figure onto itself.

5.

• \(R_{(45^{\circ},P)}\): Rotating the figure \(45^{\circ}\) about point \(P\) will not map the figure onto itself.
• \(R_{(90^{\circ},P)}\): Rotating the figure \(90^{\circ}\) about point \(P\) will not map the figure onto itself.
• \(R_{(180^{\circ},P)}\): Rotating the figure \(180^{\circ}\) about point \(P\) will map the figure onto itself.
• \(R_{(270^{\circ},P)}\): Rotating the figure \(270^{\circ}\) about point \(P\) will not map the figure onto itself.

Answer:

1. D. 5
2. \(72^{\circ}\)
3. Reflectional: Yes; Point: No
4. C. \(r_m\) and \(r_n\)
5. C. \(R_{(180^{\circ},P)}\)