Question

Circle M dilated by a scale factor of 3 gives circle N. If the circumference of circle N is 6 cm, what is the circumference of circle M?

Answer 1

For this problem, we are given two circles, M and N, that are related by a scale factor of 3, we know that the circumference of the larger circle is 6 cm, and we need to determine the circumference of the smaller one.

The circumference of a circle can be found by using the following expression:

$C=2\pi r$

Where 'r' is the radius. When the circles are scaled, their radii should be scaled at the same proportion, so the following demonstrates the relation between both radii:

$r_n=3\cdot r_m$

We know that the circumference for the circle N is equal to 6 cm, therefore we have:

$6=2\pi\cdot r_n$

If we replace the radius for circle n, with the second expression, we will obtain:

$\begin{gathered} 6=2\pi\cdot(3\cdot r_m) \\ 6=3\cdot(2\pi r_m) \\ \frac{6}{3}=C_m \\ C_m=2 \end{gathered}$

Circle M's circumference is equal to 2 cm.