Video Transcript
The following figure is an
equilateral triangle. Determine the order of rotational
symmetry of the figure. Option (A) order two, option (B)
order three, option (C) order four, option (D) order six, option (E) the figure does
not have rotational symmetry.
Let’s start by recalling that the
order of rotational symmetry of a geometric figure is the number of times you can
rotate the figure so it still looks the same as the original figure. And that’s over a rotation of 360
degrees. So let’s start by highlighting one
of the vertices on this triangle. And then we consider we turn or
rotate this triangle.
Because this is an equilateral
triangle, if we rotated this about the center, then after 120 degrees, the
highlighted vertex would now be at the base. The image after this rotation would
look the same as the original figure. If we then rotated the image
another 120 degrees or considered it as the original shape rotated 240 degrees, then
the highlighted vertex would now be on the bottom left. So the image looks like the
original shape. After a further 120 degrees of the
last image or a complete 360-degree rotation, then the shape would be back to the
original starting point.
To find the order of rotation, we
need to count how many times in this 360-degree rotation the image looked the same
as the original shape. So the image looked like itself
once after a 120-degree rotation, twice after 240 degrees, and finally a third time
when it was back in the original position. So we can give the answer that the
order of rotational symmetry is order three.
But before we finish with this
question, there’s a few things to note. The only reason that this triangle
has an order of rotational symmetry of three is because it was an equilateral
triangle. If we imagine we had even an
isosceles triangle and we rotated it through 360 degrees to see how many times it
looked the same as the original figure, we would find that that only happened
once. And that’s when it’s in the
original starting position. In this case, we would say that
this isosceles triangle would have on order of rotational symmetry of order one.
Notice that that’s also what we
mean when we say that the figure doesn’t have rotational symmetry, like we had in
option (E). However, we can give our answer
here as option (B) that this equilateral triangle and any equilateral triangle will
have an order of rotational symmetry of order three.
