Question

If the circumference of circle X is 12pi and the circumference of circle Z is 6 pi then what is the scale factor from X to Z?

3:1
1:3
1:2
2:1

Answer 1

The circumference of a circle is given by the formula \( C = 2\pi r \), where \( r \) is the radius.

For circle X:
- Circumference \( C_X = 12\pi \)
- We can find the radius \( r_X \):
\[
2\pi r_X = 12\pi
\]
Dividing both sides by \( 2\pi \):
\[
r_X = \frac{12\pi}{2\pi} = 6
\]

For circle Z:
- Circumference \( C_Z = 6\pi \)
- We can find the radius \( r_Z \):
\[
2\pi r_Z = 6\pi
\]
Dividing both sides by \( 2\pi \):
\[
r_Z = \frac{6\pi}{2\pi} = 3
\]

Now we find the scale factor from circle X to circle Z. The scale factor can be calculated using the ratio of the radii of the two circles:
\[
\text{Scale Factor} = \frac{r_Z}{r_X} = \frac{3}{6} = \frac{1}{2}
\]

Thus, the scale factor from circle X to circle Z is \( 1:2 \).

So the answer is \( \boxed{1:2} \).