What kind of figure has a rotational symmetry of 90^(∘)?
D
d
What kind of figure has a rotational symmetry of 180^(∘)?
A
a
Example Solution: (- 0.5,1), (0.5,1), (0.5,- 1), (- 0.5,- 1).
B
b
Example Solution: (- 1,0.5), (1,0.5), (1,- 0.5), (- 0.5,- 1).
C
c
Example Solution: (- 1,1), (1,1), (1,- 1), (- 1,- 1).
D
d
Example Solution: (- 1,1), (1,1), (1,- 1), (- 1,- 1).
Practice makes perfect
a
When we reflect a figure in the y-axis, the vertices of the figure change in the following way:
preimage (a,b)→ image (- a, b),
We are told that reflecting ABCD in the y-axis maps it onto itself. This means that we need one pair of vertices on the left side of the y-axis and another pair on the right side. When reflecting the figure in the y-axis, the two points on the left side should take the place of the points on the right side and vice verse.
b
When we reflect a figure in the x-axis, the vertices of the figure change in the following way:
preimage (a,b)→ image (a,- b).
We are told that reflecting ABCD in the x-axis maps it onto itself. This means that we need one pair of vertices above the x-axis and another pair below it. When reflecting the figure in the x-axis, the two points above the x-axis should take the place of the points below and vice verse.
c
If rotating a figure about the origin by 90^(∘) makes it map onto itself, it means that the quadrilateral has a 90^(∘) rotational symmetry. The only quadrilateral with this property is a square. Note that a 90^(∘) rotation about the origin changes the vertices of a figure in the following way
preimage (a,b)→ image (- b,a).
If we use this rule on, for example, A(1,1), we get a new ordered pair of B(- 1,1). Using the same rule on this new ordered pair, we get a third vertex C(- 1,- 1). Finally, we will use the rule one last time on the third vertex to get our fourth ordered pair D(1,- 1). Now we can draw our quadrilateral.
d
If rotating a figure by 180^(∘) about the origin makes it map onto itself, it means that the quadrilateral has a 180^(∘) rotational symmetry. This is the same thing as rotating the figure twice by 90^(∘). A quadrilateral with this property is a square. This means we can reuse the square we presented in part C.
