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.
