Lesson Video: Combining Transformations Mathematics

Video Transcript

In this video, we will learn how to carry out and describe combinations of transformations. The four different types of transformation are translation, reflection, rotation, and dilation. So, let’s begin by recalling what each of these transformations would look like.

The first transformation we can look at is translation. Translation is when we move a shape, but its size, shape, and orientation won’t change. If, for example, we translated this triangle five units right and two units down, then the image would appear here. Image is the word that we give for a shape after a transformation. To perform a translation, we take each of the vertices. Here, we were asked to translate five units to the right and two units down. So, doing this for each vertex gives us the location for each of the new vertices.

If we’re looking at a shape and its image and we don’t know which transformation it is, we can spot a translation because the object and its image will be the same shape and size. And notably, translations will always produce an object and its image in the same orientation. And then to describe a translation, we would need to say how the shape has moved, either in the form of words — for example, five units right and two units down — or in the form of notation, such as vector notation.

And the second type of transformation is reflection, which is when we flip or reflect an object across a line. For example, if we reflected this rectangle in the 𝑦-axis, then the image would appear on the other side. So, if we’re looking at a transformation and we’re trying to establish which one it is, a reflected object and its image will be the same size, but one will look like a reflected or mirror image of the other. When describing a reflection, we need to state the line of reflection. In our example, this would’ve been the 𝑦-axis.

Next step, let’s have a look at rotation, which is when we turn an object about a point or center of rotation. For example, if we wanted to rotate our blue triangle 90 degrees clockwise about the point marked 𝐴, then the image would appear as shown. To identify a rotation, the object and its image would be the same size, but the image would look rotated. To describe a rotation, we need to say a few more things. We need to state the center of rotation about which the object is rotated. This can either be by describing a point denoted by a letter or by using a coordinate in a grid. We also need to give the angle of rotation and the direction, for example, clockwise or counterclockwise.

The final transformation is dilation, sometimes referred to as enlargement. This is when we expand or contract an object by a given scale factor. For example, if we take our blue triangle and dilate it by scale factor two about this point 𝐴, then the image would look as shown. We can see that each of the lengths on the image are twice as long as they are on the original shape. We can easily spot a dilation because the object and its image are different size, unless the scale factor is one or negative one. In a dilation, the object and its image are usually the same orientation. But be careful as negative scale factors often make a shape look like it’s been flipped or rotated. To describe a dilation, we need to give the center of dilation and the scale factor, remembering that scale factors can be fractional and negative values too.

So, when it comes to combining transformations, that means we perform one transformation and then another transformation and maybe even another one after that. So, if we’re carrying out a transformation on a shape and we find the image after this transformation, then one of the things we need to be careful of when we’re carrying out a second transformation is whether this transformation comes from transforming the image or from transforming the original shape. We’ll now look at some questions on combining transformations. In the first question, we’ll actually carry out these transformations.

𝐴𝐵𝐶𝐷 is reflected in the 𝑥-axis and then translated five units to the right. What is the image of point 𝐵?

Here, we have a combination of transformations, firstly reflection and then translation. We should always perform them in the order in which we’re given. We’ll, therefore, begin with reflection. And we’re told that the line of reflection is the 𝑥-axis. Starting with vertex 𝐵, we can see that this is six units up from the 𝑥-axis. So, when we reflect vertex 𝐵, it will also be six units away from the 𝑥-axis but this time in the negative direction. We can call the reflected vertex 𝐵 prime.

Vertex 𝐴 is seven units away from the 𝑥-axis. So, when we reflect it, it will also be seven units away but in the negative direction. And we can create the reflected vertex 𝐴 prime. Vertices 𝐶 and 𝐷 can be reflected in the same way, giving us the images of 𝐶 prime and 𝐷 prime. We can then join the vertices to give the image. Notice that our two shapes 𝐴𝐵𝐶𝐷 and 𝐴 prime 𝐵 prime 𝐶 prime 𝐷 prime are the same size, but they look reflected.

The second transformation that we need to carry out is the translation. We’re told that 𝐴𝐵𝐶𝐷 is reflected and then translated. So, we’ll be carrying out the translation on the image rather than on the original shape 𝐴𝐵𝐶𝐷. Each vertex will be translated or moved five units to the right. So, let’s start with 𝐶 prime. If we count five hops or jumps to the right, then we can see that the image of 𝐶 prime, written as 𝐶 double prime, will lie as shown. We can find the images of 𝐴 prime and 𝐷 prime. And the image of 𝐵 prime will be 𝐵 double prime, and we note that it lies on top of 𝐶 prime. We have then created a second image.

We were asked for the image of point 𝐵 after these two transformations. So, we can give negative two, negative six as our answer. In this type of question, we could’ve simply transformed point 𝐵. But often, it’s helpful to draw the whole shape as a check that we’ve performed the transformations correctly.

We’ll now look at a question where we need to describe a combination of transformations.

The triangle 𝐴𝐵𝐶 has been transformed onto triangle 𝐴 prime 𝐵 prime 𝐶 prime which has been transformed onto triangle 𝐴 double prime 𝐵 double prime 𝐶 double prime. Describe the single transformation that maps 𝐴𝐵𝐶 to 𝐴 prime 𝐵 prime 𝐶 prime. Describe the single transformation that maps 𝐴 prime 𝐵 prime 𝐶 prime to 𝐴 double prime 𝐵 double prime 𝐶 double prime. Hence, are triangles 𝐴𝐵𝐶 and 𝐴 double prime 𝐵 double prime 𝐶 double prime similar?

In this question, we have a series or combinations of transformations which begin with triangle 𝐴𝐵𝐶. The first transformation takes us to the second smaller triangle. And the second transformation takes us to the larger triangle of 𝐴 double prime 𝐵 double prime 𝐶 double prime. We can begin by finding the first transformation between the two smaller triangles.

We can recall that the four types of transformation are translation, reflection, rotation, and dilation. If we look at the two triangles 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime, we can see that they’re the same size. Therefore, this is unlikely to be a dilation, as this usually changes the size. We can see that our two triangles are at different orientation. So, we can rule out translation as this moves the shape but keeps it the same way up.

The two triangles are not a mirror image of each other. So, we can rule out reflection. Let’s see if we could describe this transformation as a rotation. Starting with triangle 𝐴𝐵𝐶, if we rotated this in a clockwise direction, we could then work out the angles for which this must be rotated. Between 𝐴 and 𝐴 prime, there’s a right angle of 90 degrees. Between 𝐶 and 𝐶 prime, we can also see a 90-degree angle. This will confirm that we have a rotation of 90 degrees.

Notice that we’ve found this rotation by moving our vertices through the same point or coordinate. This will be the center of rotation. So, in order to fully describe this transformation, we need to put together the facts that we’ve discovered — the center of rotation, the angle, and the direction. We can then give our answer to the first part as a 90-degrees clockwise rotation about the origin. We could, of course, also have described this as a 270-degree counterclockwise rotation about the origin. Giving the coordinate zero, zero instead of the origin would also have been valid.

Let’s look at the second question. Describe the single transformation that maps 𝐴 prime 𝐵 prime 𝐶 prime to 𝐴 double prime 𝐵 double prime 𝐶 double prime. We need to be careful that we’re using the second smaller triangle and asking how we go from this to the larger triangle. If we look at our list of possible transformations, the first three — translation, reflection, and rotation — keep the object and its image the same size. As the triangles here are different sizes, that means there’s just one possible transformation, dilation.

To describe a dilation, we need to find the center of dilation and the scale factor. We can find the scale factor relatively easily by looking at how the length on the image have increased from the length on the original shape. We can compare the lengths of 𝐴 double prime 𝐵 double prime and 𝐴 prime 𝐵 prime. We can see that on 𝐴 prime 𝐵 prime, the length is two units, and on the top length of 𝐴 double prime 𝐵 double prime, this is four units long. So, it looks like we’ll have a scale factor of two. But it’s always worth checking some of the lengths on the other sides, just to be sure. The length 𝐵 prime 𝐶 prime is three units long and the length 𝐵 double prime 𝐶 double prime is six units long. And as that’s twice as large, then we’ve confirmed that the scale factor is two.

There’s a nice, easy way to find the center of dilation. To do this, we create a ray between each vertex and its image. Here, we have a ray between 𝐵 prime and 𝐵 double prime and 𝐴 prime and 𝐴 double prime. We can do the same between 𝐶 prime and 𝐶 double prime. And the place where the rays converge will be the center of dilation, which once again will be the origin or the coordinate zero, zero. We put our answer into this statement form that this will be a dilation from the origin by a scale factor of two.

Our final question asks if our triangles 𝐴𝐵𝐶 and 𝐴 double prime 𝐵 double prime 𝐶 double prime are similar. We can recall that similar means the same shape, but a different size. The angles remain the same, but the sides will be in proportion. So, in our first transformation from 𝐴𝐵𝐶 to 𝐴 prime 𝐵 prime 𝐶 prime, we didn’t change the size of these triangles, which means that they are congruent. And when we transformed 𝐴 prime 𝐵 prime 𝐶 prime to 𝐴 double prime 𝐵 double prime 𝐶 double prime, the image here did get larger. So, these two triangles would not be congruent.

Each of the lengths in the image of 𝐴 double prime 𝐵 double prime 𝐶 double prime was in proportion to those in the triangles of 𝐴 prime 𝐵 prime 𝐶 prime. All corresponding pairs of angles are congruent. So, our triangles 𝐴𝐵𝐶 and 𝐴 double prime 𝐵 double prime 𝐶 double prime are similar. So, our answer for the final part of this question is: yes, these triangles are similar.

We’ll now look at one final question on describing a combination of transformations.

A triangle with vertices three, three; seven, zero; and 10, five was transformed to one, eight; five, five; and eight, 10 and then to one, eight; negative two, four; and three, one. Which of the following describes these transformations?

And we’re given four answer options. It will be very difficult to answer this question without drawing out the coordinates. So, let’s get some grid paper and draw the vertices of the original triangle. When we’re planning our graph, it can be very helpful to make sure that we have enough room to cover the highest and lowest of our 𝑥- and 𝑦-values. Here, we have our first three coordinates drawn for the original triangle. We can then draw the second set of three coordinates to find the image of the triangle, which will look like this.

We now need to work out which transformation will take the original blue triangle to the triangle drawn in pink. We can recall that the four types of transformations are translation, reflection, rotation, and dilation. Looking at our two triangles, we can see that these are the same size. So, we can eliminate dilation as a transformation option. In a dilation, the shape and its image are usually a different size. In the diagram, we can see that the two triangles are in the same orientation. In other words, they’re not a reflection or mirror image, and they’re not a rotation of each other.

This leaves us with translation, which is when we simply move a shape to the right or left and up or down. In order to work out how a shape has been translated, we compare the vertices. Looking at the top vertex of each triangle and beginning with the blue triangle, we could see that there is a movement of two units to the left and five units upwards. We must be very careful to work with the original triangle and the image. Otherwise, instead of correctly describing this as a translation two units to the left and five units upwards, we could incorrectly describe it as a translation of two units to the right and five units down. But we can make a note that the first transformation will be a translation of two units left and five units up.

We can now look at the final set of three coordinates to find the second transformation. Plotting the coordinates one, eight; negative two, four; and three, one will give us a triangle that looks like this. We need to work out the transformation that takes our triangle in pink to the triangle in orange. Looking at our four transformation options, we can eliminate dilation as these triangles are the same size. We can also eliminate another translation as these two triangles are in a different orientation. The triangles are not a reflected or mirror image of each other. So we can eliminate reflection, which leaves us with rotation.

In order to describe a rotation, we need to give the center of rotation, the angle, and the direction. Sometimes, it can be difficult to find the center of rotation. What we’re really looking for is a point about which the shapes can pivot or turn. If we were to draw the pink triangle on tracing paper and then put the point of our pencil on this green 𝑥. Then moving our tracing paper in the direction of the green arrow would take us to the orange triangle. This is the clockwise direction.

In order to find the angle of rotation, we look at the angle between each vertex and its image. Here, we can see that we have a right angle. So, the angle of rotation will be 90 degrees. Putting these pieces of information together — where the center of rotation is at one, eight, the angle is 90 degrees, and the direction is clockwise — we would have a rotation of 90 degrees clockwise about one, eight. We could also describe this transformation as a rotation of 270 degrees counterclockwise about one, eight.

So if we look at our answer options, the first transformation was a translation of two units left and five units up. So, we can eliminate options (A) and (C). Our second transformation is a rotation of 90 degrees clockwise. So, we can eliminate option (B) as that has a rotation of 180 degrees. We can see the fully correct answer is given in option (D). It was translated two units left and five units up. And then it was rotated 90 degrees clockwise about the point one, eight.

Now, let’s summarize what we learnt in this video. We saw that there are four transformations: translation, reflection, rotation, and dilation. We can combine the transformations by performing one transformation and then another. And finally, if we’re performing or describing a series or combination of transformations, we must be careful that this is done in the correct given order.