Question
You are given the rotation r(45°, P). How many times does the rotation need to be applied to a figure to map the figure to itself?
times rotation must be applied =
times rotation must be applied =
Answers
GPT-4o mini
To determine how many times the rotation \( r(45^\circ) \) needs to be applied to map a figure to itself, we need to find the smallest integer \( n \) such that applying the rotation \( n \) times results in a total rotation that is a multiple of \( 360^\circ \).
When you apply the rotation \( n \) times, the total rotation is given by:
\[
n \times 45^\circ
\]
We want this to be a multiple of \( 360^\circ \):
\[
n \times 45^\circ = k \times 360^\circ \quad \text{for some integer } k
\]
This simplifies to:
\[
n \times 45 = k \times 360
\]
Dividing both sides by \( 45 \) gives:
\[
n = k \times 8
\]
This means that \( n \) must be a multiple of \( 8 \).
The smallest positive integer \( n \) that satisfies this condition is \( n = 8 \).
Thus, the answer is:
\[
\text{times rotation must be applied} = 8
\]
When you apply the rotation \( n \) times, the total rotation is given by:
\[
n \times 45^\circ
\]
We want this to be a multiple of \( 360^\circ \):
\[
n \times 45^\circ = k \times 360^\circ \quad \text{for some integer } k
\]
This simplifies to:
\[
n \times 45 = k \times 360
\]
Dividing both sides by \( 45 \) gives:
\[
n = k \times 8
\]
This means that \( n \) must be a multiple of \( 8 \).
The smallest positive integer \( n \) that satisfies this condition is \( n = 8 \).
Thus, the answer is:
\[
\text{times rotation must be applied} = 8
\]