Video: Dilating Shapes From a Point by Positive Scale Factors

Dilate triangle 𝐴𝐵𝐶 from the point (5, 6) by a scale factor of 2, and state the coordinates of the image.

03:03

Video Transcript

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.

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