Determine whether the following
statement is true or false. If a figure has one horizontal and
one vertical line of symmetry, then it also has rotational symmetry.
To consider what we mean by both of
these types of symmetry, let’s take an example. Here we have a square. A square has one horizontal and one
vertical line of symmetry. It does have some others as well,
but let’s just consider these two. We say that a shape has rotational
symmetry if the shape appears unchanged after rotation by an angle strictly between
zero degrees and 360 degrees. In other words, if we cut out this
square, for example, if we had it made out of cookie dough and we cut it out with a
cookie cutter, and we then turned the square through 90 degrees, it would still fit
into the original shape. Because it does that after an angle
of rotation of 90 degrees or 180 degrees, rather than just in its original position,
then this square has rotational symmetry.
So for the square that we’ve drawn
then, the statement is true. It has got the horizontal and
vertical lines of symmetry, and it has rotational symmetry. Let’s see if we can create an
interesting geometric figure based on the lines of symmetry and see what happens
with the rotational symmetry. Let’s take our two lines of
symmetry and draw a shape in one of these corners.
Because of the horizontal line of
symmetry, our shape would look like this. And then let’s consider the
vertical line of symmetry, which gives us the completed shape. If we rotated this figure through
an angle of 180 degrees, it would fit upon itself. We know that it would be upside
down, but it appears unchanged after rotation by 180 degrees. Therefore, it has rotational
We can use this pictorial example
to show that for any shape that has a horizontal and a vertical line of symmetry,
then it must also have rotational symmetry. So the answer would be true. Be careful, however. If a figure just has one line of
symmetry, then we can’t say that it also has rotational symmetry.
Before we look at the final
question, we need to cover another important concept, the angle of rotation. The angle of rotation is the
smallest angle for which the figure can be rotated to coincide with itself. Let’s recap a few geometric figures
and their orders of rotational symmetry.
A rectangle has an order of
rotational symmetry of order two, an equilateral triangle of order three, and a
square of order four. If we’re looking for the smallest
angle for which the shape can be rotated and fit upon itself, for the rectangle,
that would be 180 degrees, the equilateral triangle 120 degrees, and the square
would be 90 degrees. You may have noticed that there is
a relationship between the angle of rotation and the order of rotational
symmetry. The angle of rotation is equal to
360 degrees divided by the order of rotational symmetry. For the rectangle, 360 degrees
divided by two give us 180 degrees. For the equilateral triangle, 360
degrees divided by three gives us 120 degrees. And for the square, 360 degrees
divided by four would give us 90 degrees. Let’s see how we can apply this in
the next question.