Question

Given two concentric circles (circles that have the same center point), with

the inner circle having a radius of r1 and the outer one having a radius of

r2 as shown, give a dilation that would map the inner circle onto the outer

circle. Specify both the center of dilation and the scale factor.

Answer 1

The center of dilation has a coordinate of $(x,y) = (h,k)$ and the scale factor is $r$. The transformation formula is $r^{2} = \frac{(x_{2}-h)^{2}+(y_{2}-k)^{2}}{(x_{1}-h)^{2}+(y_{1}-k)^{2}}$.

How to define a dilation between two concentric circles

Two circles are concentric when they share the same center ($h, k$) but they have different radii ($r_{1}, r_{2}$), where $r_{1}$ corresponds to the radius of the smaller circle. By analytic geometry we know that circles are modeled after this formula:

$(x-h)^{2}+(y-k)^{2} = r^{2}$     (1)

If we assume that the center of dilation is the center of both circles and the dilation use a scale factor of $r$. Then, we have the following expression:

$r= \frac{r_{2}}{r_{1}}$    

$r^{2} = \frac{(x_{2}-h)^{2}+(y_{2}-k)^{2}}{(x_{1}-h)^{2}+(y_{1}-k)^{2}}$     (2)

The center of dilation has a coordinate of $(x,y) = (h,k)$ and the scale factor is $r$. The transformation formula is $r^{2} = \frac{(x_{2}-h)^{2}+(y_{2}-k)^{2}}{(x_{1}-h)^{2}+(y_{1}-k)^{2}}$. $\blacksquare$

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