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Geometry /

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Circle Transformation

In the diagram below, can the larger circle be mapped onto the smaller circle? What transformations are involved?

Illustration

Dilating Concentric Circles

Dilate the concentric circles below so that they map onto each other.

Example

Dilating a Small Circle to Map Onto a Large Circle

Given two concentric circles on a coordinate plane, Zosia is trying to determine the transformation that maps the smaller circle onto the larger circle.

So far, Zosia is considering four different options. Help Zosia make up her mind!

Hint

Consider the fact that the circles have different radii.

Solution

Note that the circles have different radii. The radius of the smaller circle is $3,$ and the radius of the larger circle is $6 .$

Therefore, to map the smaller circle onto the larger circle, a dilation must be performed. Furthermore, since the radius of the larger circle is twice the radius of the smaller circle, the scale factor of the dilation must be

2 .

Finally, since the circles are concentric, the center of dilation is the center of the circles, which is the point

(2, - 1) .

Example

Dilating a Large Circle to Map Onto a Small Circle

Now, given a different pair of concentric circles, Zosia wants to find the transformation that maps the larger circle onto the smaller circle.

Again, Zosia is considering four different options. Help Zosia make up her mind!

Hint

Consider the fact that the circles have different radii.

Solution

It can be seen in the diagram that the radius of the smaller circle is $3,$ and the radius of the larger circle is $9 .$

Therefore, to map the larger circle onto the smaller circle, a dilation must be performed. Furthermore, since the radius of the smaller circle is one third the radius of the larger circle, the scale factor of the dilation must be

\frac{1}{3} .

Finally, since the circles are concentric, the center of dilation is the center of the circles, which is the point

(4, 1) .

Example

Mapping Congruent Circles Using Transformations

Diego has been asked to identify the transformation that maps the circle on the left onto the circle on the right.

Diego is considering the four different options that are shown below. Which is the correct choice?

Hint

Consider the fact that the circles have the same radius.

Solution

It can be seen in the diagram that the radius of both circles is $3 .$

Therefore, to map the circle on the left onto the circle on the right, translation is the only transformation that must be performed. The circle on the left must be translated

9

units to the right and

4

units up.

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