Rotating a figure 180^(∘) is equivalent to rotating it 90^(∘) twice.
Answer: No Explanation: See solution.
Practice makes perfect
We want to decide whether it is possible for a figure to have 90^(∘) rotational symmetry but not 180^(∘) rotational symmetry. Note that rotating a figure by 180^(∘) is the same thing as rotating it by 90^(∘) twice.
If a figure has 90^(∘) rotational symmetry, each of the 90^(∘) rotations maps the figure onto itself. Therefore, a 180^(∘) rotation should map the figure onto itself as well. To confirm thi, let's consider a square, which is an example of a figure that has 90^(∘) rotational symmetry. A rotation by 90^(∘) maps it onto itself.
Rotating it 90^(∘) one more time will map onto itself again.
As we already noted, rotating a figure twice by 90^(∘) is the same as rotating it by 180^(∘). This means that, if rotating a figure 90^(∘) maps it onto itself, rotating it by 180^(∘) will map it onto itself as well. Therefore, it is impossible for a figure to have 90^(∘) rotational symmetry but not 180^(∘) rotational symmetry.
