Video: Finding the Scale Factor of a Dilation between Two Circles

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

01:49

Video Transcript

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.

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