Lesson Video: Rotations about a Point Mathematics • 8th Grade

In this video, we will learn how to identify the rotation of a shape about a point and deduce the image of a polygon after rotating about one of its vertices.


Video Transcript

In this video, we will learn how to identify rotation of a shape about a point and how to deduce the image of a polygon after a rotation about one of its vertices.

Let’s begin by recapping the different transformations. And this will allow us to identify rotations a little more easily. Firstly, we have translations. A translation moves the shape left or right and up or down. So, for example, a translation of this shape three to the right and one down produces the following image. Next, in a reflection, this creates a mirror image of a shape. So, if we take this same blue shape and reflect it in the dotted line, the image would appear as shown.

The third transformation of enlargement, often called dilation, is when a shape gets larger or smaller, except for in the cases where the scale factor is one or negative one. So, if we take this blue shape and enlarge it by a scale factor of two about this center of enlargement, the image would look like this.

Finally, we come to the main transformation that we’re looking at in this video, rotation. Rotations turn a shape in a circular motion about a point. We can describe a rotation using an angle and a direction. The direction will either be clockwise or counterclockwise. And counterclockwise is sometimes written as anticlockwise. So, let’s say that we rotated this shape 90 degrees clockwise about this center of rotation. The image will look like this, as every vertex moves 90 degrees clockwise about the center of rotation.

We’ll focus on more detail as we go through the questions, but it’s important to note that the center of rotation can be anywhere, but in this video we’ll be looking at what happens when the center of rotation is one of the vertices of the shape. Let’s take a look at the first question, in which we’ll have to identify the rotation out of a number of different transformations.

Which of the following represents a rotation of the shaded figure?

Let’s begin by recalling that a rotation is the transformation that turns a shape in a circular motion about a point. If you find it difficult to visualize transformations or even rotations, it can be helpful to use tracing paper. Put the tracing paper over the shape and trace over the outline. As we’re thinking about a rotation, it can be helpful to put our pencil point anywhere on the tracing paper and turn it and then see what happens. Out of the options that we’re given, the option (C) produces a rotation.

Just like in the sketch, if we placed our pencil point with the tracing paper here and rotated the shaded figure 90 degrees clockwise, then we’d end up with the following image. We could’ve also rotated 270 degrees counterclockwise and achieved the same image. Therefore, option (C) is the correct answer. But let’s have a quick look at the other options.

Option (A) has two shapes which look like a mirror image of each other. Therefore, this is a reflection. The line of reflection would appear halfway between these two shapes. But it’s not a rotation, so it’s not the answer. In option (B), the image is four places to the right of the original shaded figure, indicating that this is a translation but not a rotation. In option (D), this shaded figure and the unshaded figure are not congruent or similar. Therefore, this doesn’t represent any transformation and certainly not a rotation. So, we can give the answer as option (C).

In the following question, we’ll see how directions using the compass points of north, south, west, and east can be described as rotations.

Sophia went on a hike in the woods. She was walking northeast along a trail. When she got to a fork in the trail, she started walking northwest. Fully describe the rotation she made at the fork in the trail.

In this question, we’re told that Sophia is walking through the woods. She is walking northeast. This means that we’ll need to recall our compass directions. We always have north at the top, south at the bottom, and then west and east. As Sophia is walking northeast, then she’s walking in this direction. So, here’s Sophia walking along the trail until she gets to a fork in the trail. A fork in a trail would be a place where there are a number of different paths.

We’re told that Sophia then starts to walk Northwest. Northwest is between north and west, so Sophia starts to walk in this direction. We’re then asked to describe the rotation she made at the fork in the trail. This is the point where she changed direction. When Sophia got to the fork in the trail, she was looking in the northeast direction. Instead of continuing forward, however, she turned to the northwest direction. This turn indicates a rotation. And in order to describe a rotation, we need to give the angle of the rotation and its direction.

If we look at our compass points, the angle between north and east is a right angle of 90 degrees. Therefore, the angle between north and northeast is exactly half of this. It’s 45 degrees. The same is true for the angle between north and northwest. It’s also 45 degrees. So, when Sophia turned, she turned at an angle of 90 degrees. The direction of this turn is counterclockwise. We can, therefore, give our answer that the rotation made is a rotation by 90 degrees counterclockwise.

We could also have described this as a rotation by 270 degrees clockwise. Usually when we’re describing the rotation of a shape, we’d also need to give the point about which the shape is rotated. In this case, the center of rotation is the fork in the trail, and that was given to us in the question. Therefore, an answer of rotation by 90 degrees counterclockwise is sufficient.

Let’s have a look at another question.

Where does a rotation through 180 degrees about 𝑀 send the segment 𝐹𝐴?

Let’s remember that a rotation is the transformation that turns a shape in a circular motion about a point. In this question, the shape is the line segment 𝐹𝐴 and the rotation is 180 degrees about the point 𝑀. Usually, we have a direction of motion or turn, but since 180 degrees is exactly half of a full turn of 360 degrees, then we can achieve that in either the clockwise or the counterclockwise direction. Let’s start with vertex 𝐴 and see where its image would appear after a rotation of 180 degrees.

Each vertex will always be the same distance from the center of rotation. For example, if we rotated 𝐴 60 degrees clockwise about 𝑀, then its image would appear at vertex 𝐹. We know that it would be an angle of 60 degrees that would rotate 𝐴 to 𝐹 as we have a regular hexagon. Therefore, if we rotated 180 degrees and the direction of this arrow is clockwise, then the image of 𝐴 will be at vertex 𝐷.

Next, we can consider where the image of point 𝐹 would be after a 180-degree rotation. It would be here at vertex 𝐶. Therefore, we can give our answer that a rotation of 180 degrees about 𝑀 sends the segment 𝐹𝐴 to line segment 𝐶𝐷.

In the following question, we’ll have to work out the angle of rotation between an object and its image.

Triangle 𝐴 𝐵 prime 𝐶 prime is the image of triangle 𝐴𝐵𝐶 by a counterclockwise rotation of 𝑥 degrees about 𝐴. Find 𝑥.

So, in this question, we’re given that there’s a rotation. Triangle 𝐴𝐵𝐶 is rotated, and we’re told that the direction is counterclockwise, to give us triangle 𝐴 𝐵 prime 𝐶 prime. The angle of rotation is 𝑥 degrees, and we need to work out what 𝑥 is.

The center of rotation of this rotation is at vertex 𝐴, which explains why there’s no new image of 𝐴 of 𝐴 prime. In order to work out the angle of rotation between an object and its image, we can calculate the angle between any vertex and the image of that vertex. Let’s begin with vertex 𝐵. We know that it’s rotated counterclockwise to the image 𝐵 prime. If we joined each vertex, 𝐵 and 𝐵 prime, with a line to the center of rotation, then to find the angle of rotation, we just need to find the angle between these two lines.

So, 𝑥, the angle of rotation, is equal to 37 degrees plus 69 degrees, which means that 𝑥 is equal to 106. We can check our answer by finding the angle between 𝐶 and 𝐶 prime. To find the angle here, once again we’ll be adding 69 degrees and 37 degrees, which gives us an answer of 106.

In the final question, we’ll perform a rotation on the coordinate grid.

Find the new positions of the given triangle’s vertices after rotating it 180 degrees counterclockwise about 𝐿.

On this grid, we have a triangle 𝐿𝑀𝑁. And we’re told that we need to rotate it 180 degrees counterclockwise. This means that we’ll be turning this shape. And the fact that we’re turning it about 𝐿 indicates that 𝐿 is the center of rotation. Let’s begin by looking at the rotation of vertex 𝑁. As we have a rotation of 180 degrees, we can use the fact that the angles on a straight line sum to 180 degrees to think that the new vertex of 𝑁 must lie somewhere on this straight line.

However, the distance of a vertex from the center of rotation will always stay the same. Vertex 𝑁 is one unit right and three units out from the center of rotation. So, when it’s rotated, the new vertex will be one unit to the left and three units down. We can call this image of 𝑁 𝑁 prime. Next, let’s have a look at where the vertex of 𝑀 will appear after the rotation. Turning through 180 degrees will place the image of 𝑀 here. We can call this new vertex 𝑀 prime. We can join these three vertices to create the image of the triangle 𝐿𝑀𝑁.

To answer the question, then, we need to give the three coordinates of the three new vertices. Vertex 𝐿 prime is the same as vertex 𝐿, and it’s at the point five, four. The image of vertex 𝑀 is 𝑀 prime, and it’s at the point two, three. Finally, the image of the vertex 𝑁 is at 𝑁 prime, and it’s at the position four, one. And so, we have the answer for the positions of the triangle’s vertices after a rotation. Notice that because it’s 180 degrees, it wouldn’t matter if we rotated counterclockwise or clockwise.

We can now summarize what we’ve learnt in this video. Firstly, we saw that rotation is one of the transformations. The other transformations are translation, reflection, and enlargement. We saw that a rotation turns a shape through an angle in a circular motion about a point. In this video, we focused on the occasions when the center of rotation is one of the vertices, but the center of rotation can occur anywhere inside or outside the shape.

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