Question

*84.) for each regular polygon below, list 5 different numbers of degree that would map the figure onto itself.
***#5.)
a regular decagon is rotated n degrees about its center, carrying
the decagon onto itself. the value of n could be
(1) 10° (3) 225°
(2) 150° (4) 252°
#6.) name 4 polygons that are mapped onto
themselves after a 180° rotation.
***#7.) the regular polygon below undergoes a
rotation of a° to map onto itself. which of the following
could be the value of a? select all that apply.
\\( \frac { 3 6 0 } { 8 } = 4 5 ^ { \circ } \\)
\\( n = 8 \\)
a.) 45°
b.) 60°
c.) 225°
d.) 72°
e.) 144°

Explanation:

Step1: Calculate the minimum rotation degree for a regular decagon

The formula for the minimum rotation degree \(r\) of a regular \(n -\)sided polygon is \(r=\frac{360^{\circ}}{n}\). For a decagon (\(n = 10\)), \(r=\frac{360^{\circ}}{10}=36^{\circ}\). The values of \(n\) that map the decagon onto itself are multiples of \(36^{\circ}\).

• For \(n = 10^{\circ}\), \(\frac{10}{36}=\frac{5}{18}\) (not an integer).
• For \(n = 150^{\circ}\), \(\frac{150}{36}=\frac{25}{6}\) (not an integer).
• For \(n = 225^{\circ}\), \(\frac{225}{36}=\frac{25}{4}\) (not an integer).
• For \(n = 252^{\circ}\), \(\frac{252}{36}=7\) (an integer).

Step2: Analyze the regular octagon (\(n = 8\)) rotation

The minimum rotation degree \(r=\frac{360^{\circ}}{8}=45^{\circ}\). The values of \(a\) that map the octagon onto itself are multiples of \(45^{\circ}\).

• For \(a = 45^{\circ}\), \(\frac{45}{45}=1\) (an integer).
• For \(a = 60^{\circ}\), \(\frac{60}{45}=\frac{4}{3}\) (not an integer).
• For \(a = 225^{\circ}\), \(\frac{225}{45}=5\) (an integer).
• For \(a = 72^{\circ}\), \(\frac{72}{45}=\frac{8}{5}\) (not an integer).
• For \(a = 144^{\circ}\), \(\frac{144}{45}=\frac{16}{5}\) (not an integer).

Answer:

For #5: (4) \(252^{\circ}\)
For #7: a. \(45^{\circ}\), c. \(225^{\circ}\)