Question

if the circumference of circle x is 12pi and the circumference of circle z is 6pi, then what is the scale factor from x to z

Answer 1

To determine the scale factor from circle x to circle z based on their circumferences, we can use the relationship between their circumferences.

The formula for the circumference \( C \) of a circle is given by:

\[
C = 2\pi r
\]

Where \( r \) is the radius of the circle.

1. For circle x, the circumference is \( 12\pi \). We can set up the equation:

\[
12\pi = 2\pi r_x
\]

Dividing both sides by \( 2\pi \):

\[
r_x = \frac{12\pi}{2\pi} = 6
\]

2. For circle z, the circumference is \( 6\pi \). Setting up the equation:

\[
6\pi = 2\pi r_z
\]

Dividing both sides by \( 2\pi \):

\[
r_z = \frac{6\pi}{2\pi} = 3
\]

3. Now that we have the radii of both circles, we can find the scale factor from circle x to circle z. The scale factor is given by the ratio of the radii:

\[
\text{Scale factor} = \frac{r_z}{r_x} = \frac{3}{6} = \frac{1}{2}
\]

Thus, the scale factor from circle x to circle z is \( \frac{1}{2} \).