Video: Rotations on the Coordinate Plane

In this video, we will learn how to find the vertices of a shape after it undergoes a rotation of 90, 180, or 270 degrees about the origin clockwise and counterclockwise.

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Video Transcript

In this video, we will learn how to find the vertices of a shape after it undergoes a rotation of 90 degrees, 180 degrees, or 270 degrees clockwise or counterclockwise about the origin. We will also find out how to describe a rotation.

Let’s begin by recalling that a rotation is a type of transformation that turns a figure around a point or the center of rotation. We’ll also need to cover some important things. Let’s start with angles. In this video, we’ll be looking at rotations with angles of 90 degrees, 180 degrees, and 270 degrees. A 90-degree angle is a right angle. A 180-degree angle is the type of angle you would find on a straight line. And a 270-degree angle would look like this. It can also be helpful to remember that this other angle, created from a 270-degree angle, is a right angle of 90 degrees.

When discussing rotations, it’s also helpful to review the two different directions that we use. These are clockwise and counterclockwise. Sometimes we may see counterclockwise referred to as anticlockwise. The clockwise direction will be in the same direction as the hands move on a clock. And a counterclockwise turn would be in the opposite direction.

The center of rotation is very important when we’re discussing rotations as it will determine where the shape is positioned. For example, if we take this rectangle and rotate it 90 degrees clockwise about the center of rotation marked 𝑥, then the image of the shape, that’s the shape after it’s being rotated, would look like this. If instead we rotated this rectangle 90 degrees clockwise and we put the center of the rotation on one of its vertices, then the image would look like this. If we had the center of rotation within the rectangle, then the image would look like this. And that’s why it’s important, as we’ll see later in this video, not only to describe the angle and the direction, but also to say where the center of rotation would be.

And before we begin some questions. Here’s a helpful tip if you’re having difficulties with rotations. Using tracing paper can help us see where a shape should be positioned after rotation. A good backup if we don’t have tracing paper is baking or parchment paper, the kind we might use to stop pastries sticking to the tray in the oven. We put the tracing or parchment paper over the drawing, trace around the shape. And then, for example, if we needed to rotate by 180 degrees, we would put the point of our pencil into the center of rotation and rotate the tracing paper around. We would then be able to see where the rotated shape would appear. So now, let’s look at some questions where we need to carry out a rotation.

Determine the coordinates of the vertices’ images of triangle 𝐴𝐵𝐶 after a counterclockwise rotation of 180 degrees around the origin.

In this question, we’ll be performing a rotation. So we’ll be turning the shape. We’ll be rotating this triangle through an angle of 180 degrees. And we’re told to do this in a counterclockwise direction, although, for a 180-degree angle, it doesn’t matter whether the direction is clockwise or counterclockwise. The center of rotation here is the origin. That’s the coordinate zero, zero. So let’s go ahead and carry out this rotation.

Beginning with point 𝐴, we can see that this is located with an 𝑥-coordinate of negative eight and a 𝑦-coordinate of seven. So when we rotate this 180 degrees counterclockwise, the 𝑥-coordinate will be at eight and the 𝑦-coordinate will be at negative seven. In other words, there will still be a movement of eight units in the 𝑥-direction. It was eight units to the left or negative eight. And it’s now eight units to the right or positive eight. The 𝑦-coordinate will still be seven units away. It was positive seven units and it’s now negative seven units. We could label the image of vertex 𝐴 as 𝐴 prime.

Vertex 𝐵 on the triangle is at negative three, seven. So when we rotate it 180 degrees, it will still have an 𝑥-coordinate value of three, this time of positive three, and at a distance of seven in the 𝑦-axis, this time at negative seven. And we can label this new vertex as 𝐵 prime. As a check, we can notice that the line joining 𝐴 prime and 𝐵 prime is also horizontal in the same way as the line joining 𝐴𝐵. This is because our horizontal line which has been rotated 180 degrees would also produce another horizontal line. Our final vertex, 𝐶, is at negative four, three. So its image, 𝐶 prime, will be at the coordinate four, negative three. And we could complete the drawing of the triangle 𝐴 prime 𝐵 prime 𝐶 prime.

In a rotation, the object and its image will always stay the same size. So it’s worth checking some key lengths to see if they’re the same size as in the original object and its image. We can see that both the original shape and its image have a horizontal length of five units. And both triangles have a perpendicular height of four units. The question asked us to write the coordinates of the vertices’ images. So we can write 𝐴 prime as eight, negative seven, 𝐵 prime as three, negative seven, and 𝐶 prime is four, negative three. We can also see in this question that, in a rotation of 180 degrees about the origin, a point 𝐴 with coordinate 𝑥, 𝑦 will be rotated to give the image 𝐴 prime of coordinates negative 𝑥, negative 𝑦.

If we look at the original vertex 𝐴 with coordinate negative eight, seven, the image 𝐴 prime had the coordinate eight, negative seven. In the same way, the coordinate 𝐵 at negative three, seven became 𝐵 prime at three, negative seven. Coordinate 𝐶 at negative four, three became 𝐶 prime at four, negative three. After a rotation of 180 degrees about the origin, if the value of the 𝑥 was positive, it becomes negative and if it was negative, it becomes positive. The same is true for the 𝑦-value. This fact can be a helpful check whenever we’re carrying out this kind of rotation. Here, we can list our final coordinates for the answer.

In the last question, we saw how a 180-degree rotation changes a point 𝐴 with coordinates 𝑥, 𝑦 into a point 𝐴 prime with coordinates negative 𝑥, negative 𝑦. We might wonder if there are similar rules for rotations of other sizes. And the answer is yes. For a 90-degrees clockwise rotation about the origin, a point 𝐴 with coordinate 𝑥, 𝑦 will have an image 𝐴 prime where the 𝑥-coordinate is the original 𝑦-value and the 𝑦-coordinate is the original negative 𝑥-value. We can also apply this rule for a 270-degree counterclockwise rotation.

There is also another rule for a 270-degree clockwise rotation. A point 𝐴 with coordinate 𝑥, 𝑦 will become 𝐴 prime where the 𝑥-coordinate is the negative 𝑦 original value and the 𝑦-coordinate is the original 𝑥-value. This rule can also be applied to a 90-degree counterclockwise rotation. It may be helpful to note down these rules as they will help us as we go through. But we do need to be careful to remember that these rotation rules only apply when the rotation is around the origin.

In the previous question, we also saw one other key point about rotations. And that is that, in a rotation, the object and its image are the same shape and size. In other words, they’re congruent. Now, let’s take a look at another rotation question.

Determine the coordinates of the vertices’ images of triangle 𝐴𝐵𝐶 after a counterclockwise rotation of 270 degrees around the origin.

In this question, we have a rotation, which means that the shape is going to turn. The point at which it’s going to turn or the center of rotation is the origin. That’s the coordinate zero, zero. The angle of rotation is 270 degrees and the direction is counterclockwise. So the angle of 270 degrees will look like this and the direction will be like this, counterclockwise. This would also be equivalent to a 90-degree clockwise turn. So a 270-degree counterclockwise rotation will send our triangle into this quadrant.

Let’s begin by looking at our point 𝐴 and establishing how far away it is from our center of rotation. It is negative seven units on the 𝑥-axis and negative three units on the 𝑦-axis. If we consider this as a 90-degree clockwise rotation, then it will still be seven units away but this time on the 𝑦-axis and three units, or rather this time negative three units, on the 𝑥-axis. We can label this new point of the image as 𝐴 prime. Point 𝐵 is at negative three on the 𝑥-axis and negative four on the 𝑦-axis. And after the rotation, it will be three units on the 𝑦-axis and negative four on the 𝑥-axis. We can label this as 𝐵 prime and create the line segment 𝐴 prime 𝐵 prime.

For our final point, 𝐶, we can see that this is at the coordinate negative six, negative five. So the rotation of this will be at six units on the 𝑦-axis and negative five on the 𝑥-axis. Once we’ve joined the vertices, it’s always worth checking that the dimensions of each of the original and the image are the same size. For example, the line 𝐴𝐶 is two units down and one unit across. And the line 𝐴 prime 𝐶 prime is also two units by one unit. We can also compare the lines 𝐵𝐶 and 𝐵 prime 𝐶 prime by noticing that they are both three units by one unit.

When answering this question, we could also have used the rule that a 270-degree counterclockwise rotation around the origin means that a point 𝐴 with coordinates 𝑥, 𝑦 will have an image 𝐴 prime with coordinates 𝑦, negative 𝑥. So if we take our vertex 𝐴 with coordinates negative seven, negative three, then to find the coordinates of the image 𝐴 prime, the 𝑥-coordinate would be the same as the original 𝑦-coordinate and to find the 𝑦-coordinate, this will be the same as the original 𝑥-coordinate, but with a switched sign. So here, 𝐴 prime is negative three, seven. And we can see from our diagram that this is indeed the coordinate of vertex 𝐴 prime.

In the same way, vertex 𝐵 with coordinate negative three, negative four becomes the vertex 𝐵 prime where the 𝑥-coordinate is the original 𝑦-coordinate and the 𝑦-value would be the negative of what was the original 𝑥-value. And we did indeed find that 𝐵 prime is at negative four, three. We can also see from using this rule and also the diagram that 𝐶 prime is at negative five, six. We can then list the coordinates of the vertices’ image 𝐴 prime, 𝐵 prime, and 𝐶 prime.

In the first three questions, we were given a rotation and asked to carry it out. Now, we’re going to look at the case where we’re given a diagram of a rotation and we need to describe it. When describing a rotation, one of the first things we need to think about is the angle through which the rotation is carried out. We’ll also need to describe the direction of the rotation. We might notice in our diagram that there is a 90-degree angle of rotation, but we’d also need to say that that was in the clockwise direction. So we would have a 90-degree clockwise rotation here. We could have also described this as a 270-degree counterclockwise rotation.

But there is one final piece of information that we also need to include when describing a rotation. And that is to state the center of rotation. In this example, the center of rotation would be here at the origin zero, zero. Putting it all together then, we could describe this as a 90-degree clockwise rotation around the origin or a 90-degree clockwise rotation about the origin. We’ll now look at a question where we describe the rotation of a point.

A triangle graphed on the coordinate plane has a vertex at six, zero. Which of the following rotations would move the vertex to the point zero, six? Option (A) 90 degrees clockwise around the origin. Option (B) 90 degrees counterclockwise around the origin. Or option (C) 180 degrees clockwise or counterclockwise around the origin.

We could start this question by sketching out where the original vertex would be. We’re told that this vertex is at six, zero. Let’s call this original vertex 𝐴. The image of this vertex after the rotation is at zero, six. Let’s call this 𝐴 prime. So how would we describe the rotation from 𝐴 to 𝐴 prime? We can recall that, to describe a rotation, we need to say three things, the angle, the direction, and the center of rotation. If we were to draw the angle between 𝐴 and 𝐴 prime, we would see that there’s a right angle of 90 degrees.

In terms of the direction, the direction that’s opposite to the one in which the hands turn on a clock is called counterclockwise, and that’s the direction that we can see here. The center of rotation will be the origin or the coordinate zero, zero. To put these three pieces of information together, we would say that this is a rotation of 90 degrees counterclockwise around the origin. But there is one other way in which we could have performed this rotation. And that is that we could have gone from vertex 𝐴 to 𝐴 prime by going in the opposite direction through an angle of 270 degrees.

Here, the center of rotation would have stayed the same, but we would have described the rotation as a 270-degree clockwise rotation around the origin. However, out of these two descriptions, only one appears in our answer options. And that is the one given in option (B), a rotation of 90 degrees counterclockwise around the origin.

Now, let’s summarize the key points of this video. We learned that a rotation is a transformation that turns a figure around a center of rotation. The angle of rotation can be any size. In this video, we looked particularly at rotations of 90 degrees, 180 degrees, and 270 degrees. We saw that there are two directions that we use when discussing rotations, clockwise and counterclockwise. We saw that, in a rotation, the object and its image are congruent. That means they’re the same shape and size. We saw three different rules for how the coordinates of a vertex change under rotations about the origin. And finally, we saw that when we’re describing a rotation, we need to state the angle, the direction, and the center of rotation.

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