To find the scale factor from circle X to circle Z based on their circumferences, we can use the formula for the circumference of a circle, which is \(C = 2\pi r\), where \(r\) is the radius.
Given:
- Circumference of circle X: \(C_X = 12\pi\)
- Circumference of circle Z: \(C_Z = 6\pi\)
The radii of the circles can be calculated as follows:
-
For circle X: \[ C_X = 2\pi r_X \implies 12\pi = 2\pi r_X \implies r_X = \frac{12\pi}{2\pi} = 6 \]
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For circle Z: \[ C_Z = 2\pi r_Z \implies 6\pi = 2\pi r_Z \implies r_Z = \frac{6\pi}{2\pi} = 3 \]
Now, to find the scale factor from circle X to circle Z, we can compare their radii:
\[ \text{Scale factor} = \frac{r_X}{r_Z} = \frac{6}{3} = 2 \]
Therefore, the scale factor from circle X to circle Z is:
2:1 (ratio of 2 to 1).