Question Video: Relating Lines of Mirror Symmetry to Rotational Symmetry Mathematics

Determine whether the following statement is true or false: If a figure has one vertical line of symmetry, then it also has rotational symmetry.

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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.

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