Video: Dilation: Positive Scale Factors

In this video, we will learn how to dilate shapes about a center of dilation by positive integer or fractional scale factors and how to describe dilations.

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Video Transcript

In this video, we will learn how to dilate shapes about a center of dilation by positive scale factors which will be either integers or fractional values. We’ll also learn how to describe dilations by determining their scale factor and center of dilation. Dilations can also be called enlargements, so the two terms will both be used in this video.

So, what is a dilation? Well, it’s a particular type of transformation, which just means an operation which moves, turns, reflects, or changes the size of an object, usually drawn on a coordinate grid.

A dilation is one which changes the size of the object and its position. There are two key pieces of information that we need to be given in order to perform a dilation. The first of these is the scale factor. This is a number which tells us how many times bigger we need to make the shape. For example, a scale factor of two means that every length on the new shape, which we call the image, must be twice as long as the corresponding length on the original shape, which we call the object. The second key piece of information we need is the center of dilation or center of enlargement. This is the point where we are performing the dilation from. If the scale factor for the dilation is two, then every point on the image must be twice as far from this point as it was for the original object.

One way we can understand dilations is by thinking about the shadow created by a single light source. Suppose you shine a light directly at a wall and stand in front of it. Your shadow will appear on the wall, but it will be a different size depending on how close you’re standing to the light source. Here, the position of the light source is the fixed point or the center of dilation. The scale factor for the dilation could be found by dividing the height of your shadow by your true height. When we dilate an object, its image is still the same shape. And all of its interior angles remain the same. Corresponding lengths on the two shapes in the same ratio. For example, they’re all twice as long or half as long. The object and its image are therefore similar shapes. Let’s now have a look at some examples of how we can perform and describe dilations.

Dilate triangle 𝐴𝐵𝐶 from the point five, six by a scale factor of two and state the coordinates of the image.

We’ve been told in the question that we’re going to perform this dilation from the point five, six, that is, the center of dilation. And it’s this point here on the figure. The scale factor we’re using is two, which means that every point on the image of this triangle after it’s been dilated must be twice as far from the center of dilation as it was originally. Let’s start with vertex 𝐴 on the triangle. We can see that to get to vertex 𝐴 from the center of dilation, we need to move four squares down and then three squares to the left. To determine the position of this vertex on our dilated triangle, we need to double each of these distances. So, from the center of dilation, we now need to move eight units down and six units to the left, which takes us to the point with coordinates negative one, negative two.

We often label the vertices of the image using prime notation. So, the image of vertex 𝐴 is 𝐴 prime. Now, let’s consider vertex 𝐵. To get from the center of dilation to vertex 𝐵, we move two units down and no units across. Doubling this distance then because, remember, our scale factor is two, the image of vertex 𝐵 will be four units below the center of dilation. That’s the point with coordinates five, two. So, we can label this point as 𝐵 prime. Finally, we consider vertex 𝐶 which is four units to the left of the center of dilation. The image of this vertex then will be eight units to the left of the center of dilation. That’s the point with coordinates negative three, six which we can label as 𝐶 prime. By joining these three vertices together, we now have the image of triangle 𝐴𝐵𝐶, that’s 𝐴 prime 𝐵 prime 𝐶 prime, following this dilation.

We’re also asked to state the coordinates of the image. So, the image of 𝐴, 𝐴 prime, is negative one, negative two. The image of 𝐵, 𝐵 prime, is five, two. And the image of 𝐶, 𝐶 prime, is negative three, six. It’s also helpful to check some lengths on the object and the image to confirm we’ve used the correct scale factor. Now, this is slightly tricky here as none of the lengths are horizontal or vertical lines, but we could consider the line 𝐴𝐶 and the line 𝐴 prime 𝐶 prime. Looking at the line connecting 𝐴 and 𝐶 then, we see that it moves one unit to the left and four units up. On the image, the line connecting 𝐴 prime and 𝐶 prime moves two units to the left and eight units up. So, each of these distances have been doubled, which means that the length of 𝐴 prime 𝐶 prime will be twice the length of 𝐴𝐶. So, the scale factor is, indeed, two.

In our previous example, we saw that when we perform a dilation with a scale factor of two, all of the lengths are doubled. Each point on the image is twice as far from the center of dilation as it was on the object. And the image is larger than the object. We can also perform dilations where the scale factor is a fraction. But what does this mean? Well, it essentially means the reverse. All of the lengths will be halved. Each point on the image will be half as far from the center of dilation as it was on the object. And in fact, the image will be smaller than the object. Let’s now consider an example of this.

Dilate the rectangle 𝐴𝐵𝐶𝐷 from the origin by a scale factor of one-third and state the coordinates of the image.

We’re told in this question that the center of dilation is the origin. That’s the point zero, zero. It’s this point here. A scale factor of one-third, that’s a positive fraction less than one, means that the image after dilation will be smaller than the original object. And it will also be closer to the center of dilation, the origin. We should check both of these things are true once we’ve completed the problem. Let’s take each vertex of the shape in turn, starting with vertex 𝐴. To get to vertex 𝐴 from the center of dilation, we go six units to the right and three units up. The image of vertex 𝐴 following the dilation will be a third of these distances from the origin. That’s two units to the right and one unit up. So, we can plot the vertex 𝐴 prime of the image at the point with coordinates two, one.

To get to vertex 𝐵 from the center of dilation, we go three units right and three units up. The image of this vertex then will be a third of these distances from the center of dilation. That’s one unit right and one unit up. So, we can plot the vertex 𝐵 prime at the point with coordinates one, one. For the vertex 𝐶, this is three units right and nine units up from the center of dilation. Its image will therefore be one unit right and three units up from the center of dilation. So, we can plot 𝐶 prime at the point with coordinates one, three. We can work out the coordinates of 𝐷 prime in exactly the same way and then join the four vertices together to find the image 𝐴 prime 𝐵 prime 𝐶 prime 𝐷 prime.

We’re asked to state the coordinates of the image. They are two, one; one, one; one, three; and two, three. Notice that the image is indeed smaller than the object and is closer to the center of dilation. So, the two things we said we needed to check are indeed true. We can also use pairs of corresponding sides to confirm we’ve used the correct scale factor. If we divide a length on the image by the corresponding length on the object, we should get a scale factor of one-third. And indeed, this is true for the side 𝐶 prime 𝐷 prime and 𝐶𝐷. It’s also true for the sides 𝐴 prime 𝐷 prime and 𝐴𝐷. The scale factor here is two over six, which simplifies to one-third. So, we can be confident that we’ve performed this dilation correctly.

We’ve now seen how to perform a dilation using a positive integer scale factor and a positive fractional scale factor. In our next two examples, we’ll see how to determine the scale factor and center of dilation when we’ve been given a diagram of an object and its image.

Circle 𝐵 is a dilation of circle 𝐴. What is the scale factor of the dilation?

To answer this problem, we remember that the scale factor for a dilation can be calculated by comparing corresponding lengths on the object and its image. We can divide any length on the image, which we can think of as a new length, by the corresponding length on the object, which we can think of as an original length. We need to be careful we get the object and its image the right way around. We’re told that circle 𝐵 is a dilation of circle 𝐴, which means that circle 𝐴 is the original object and circle 𝐵 is the image. Perhaps we could consider then the diameters of these circles. On circle 𝐴, we can see that the diameter is six units. That’s the difference between nine and three units. And on circle 𝐵, the diameter is four units. That’s the difference between two and negative two.

So, dividing the new length, that’s the length on the image, by the original length, that’s the length on the object, we have a scale factor of four-sixths. Of course, this can and should be simplified to two-thirds. Now, notice that this is a positive fraction less than one, which means that the image should be smaller than the object. And we can see that this is indeed true on our diagram. So, it gives us some confidence that we’ve performed this division the correct way around. We could also have answered this question by considering the radii of the two circles. Here, I’ve used the vertical radii. And if we did, we would once again have arrived to an answer of two-thirds. So, by comparing corresponding lengths on the two shapes, we found that the scale factor is two-thirds.

In our final example, we’ll see how we can use the position of an object and its image on a coordinate grid to determine the coordinates of the center of dilation.

Triangle 𝐴 prime 𝐵 prime 𝐶 prime is the image of triangle 𝐴𝐵𝐶 following a dilation with a scale factor of two. Determine the coordinates of the center of dilation.

So, in this question, we’ve been given the scale factor of the dilation, it’s two, but we don’t know the center of dilation, the point from which the dilation has occurred. As the scale factor of the dilation is two, then this means that each vertex of triangle 𝐴 prime 𝐵 prime 𝐶 prime must be twice as far from the center of dilation as the corresponding vertex on triangle 𝐴𝐵𝐶. This means that the center of dilation must be somewhere in this region here. But we don’t need to use trial and error to find it. Thinking back to our explanation of dilations using the light source, we know that we can draw straight lines from the center of dilation which pass through corresponding vertices on the object and its image. These lines are sometimes called rays. And we can use these lines to work backwards and determine the center of dilation.

First, we draw a straight line which connects vertex 𝐴 and 𝐴 prime. Then, we draw a straight line connecting vertices 𝐵 and 𝐵 prime. And finally, a straight line connecting vertices 𝐶 and 𝐶 prime. We should find, and indeed we do, that these three lines all intersect at a single point. And this point is the center of dilation. The coordinates of this point on our diagram are negative three, zero. Now, in fact, we only needed to draw two of these lines. But by drawing all three and confirming that they do indeed all intersect at the one single point, we’ve got a little bit more confidence in our answer. We can also see from the figure that vertex 𝐶, for example, is two units right and one unit up from this point. And vertex 𝐶 prime is four units right and two units up from this point. So, the scale factor of the dilation is indeed two. The coordinates of the center of dilation are negative three, zero.

Let’s now review some of the key points that we’ve seen in this video. Firstly, a dilation is a transformation which changes the size and position of an object. A dilation is described by two key pieces of information: firstly, its scale factor, and secondly, its center. The object and its image are similar shapes. And to find the scale factor of a dilation, we can divide any length on the image by the corresponding length on the object, which we can think of as new length divided by original length. If the scale factor of a dilation is greater than one, then the dilated shape will be larger. Whereas if the scale factor is a positive fraction between zero and one, then the dilated shape will be smaller. Finally, to find the center of a dilation, we can draw lines or rays connecting corresponding vertices on the object and its image and look for their common point of intersection.

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