Video Transcript
Determine whether the following
statement is true or false. If a figure has one vertical line
of symmetry, then it also has rotational symmetry.
In this question, we need to recall
two different types of symmetry: firstly line symmetry, demonstrated through
reflection, and secondly rotational symmetry. We say that a shape has rotational
symmetry if the shape appears unchanged after a rotation about a point by an angle
whose measure is strictly between zero degrees and 360 degrees. That essentially means if we turn a
shape through 360 degrees and the shape after rotation looks the same as it did when
we started, then it has rotational symmetry.
Notice that we don’t include the
angle of zero degrees or 360 degrees as that would just be the starting
position. Perhaps the best way to answer a
question like this is to try drawing a few shapes that do have a line of symmetry
and see if they do or don’t also have rotational symmetry. So let’s pick a rectangle to begin
with.
The shape in question has to have a
vertical line of symmetry, and the rectangle does. Let’s then imagine that we have a
center of rotation in the center of the rectangle. And we begin to rotate the
rectangle. After 90 degrees, the rectangle
would look like this in green. But it doesn’t look the same as it
did to start off with.
Let’s continue to turn the shape
through another 90 degrees. Now, we can see that the rotation
sits on top of itself. The rectangle is effectively upside
down. But the original shape would look
the same as this rotated image. So the shape we started with did
have a vertical line of symmetry. And because it fits upon itself
after 180 degrees, we say that it does have rotational symmetry. This would be a good example to
show that the statement is true.
But let’s see if we can disprove it
and show it’s false. Let’s take this figure. It’s actually an isosceles
trapezoid because it’s got a pair of parallel sides and the two nonparallel sides
are equal in length. And that gives us this vertical
line of symmetry. Let’s consider what happens if we
rotate this shape through up to 360 degrees. Well, there are no points where
this trapezoid will look the same other than the original starting position. Even, for example, after a
180-degree rotation, the trapezoid would look upside down. That is, the base, which is longer,
will be at the top of the figure, which means that this shape would not have
rotational symmetry.
Considering the statement in the
question then — if a figure has one vertical line of symmetry, then it also has
rotational symmetry — we can give the answer as false. We found one occasion where it was
true, but we found on occasion where it’s false. Therefore, this statement will not
always be true.
