To prove that two circles are similar, we need to show that they have the same shape, which is true if their corresponding parts are in proportion. In the case of circles, the defining characteristic is their radius.
Let's analyze the given circles:
-
Circle X:
- Center: \((-2, 8)\)
- Radius: \(6\)
-
Circle Y:
- Center: \((4, 2)\)
- Radius: \(3\)
Step 1: Calculate the ratio of the radii.
- The radius of Circle X is \(6\).
- The radius of Circle Y is \(3\).
The ratio of the radii of Circle X to Circle Y is: \[ \text{Ratio} = \frac{\text{Radius of Circle X}}{\text{Radius of Circle Y}} = \frac{6}{3} = 2 \]
Step 2: Check the definition of similarity for circles. Two circles are similar if the ratio of their corresponding radii is constant (in this case, it's \(2\)). This means that all corresponding linear dimensions of these circles (which here is just the radius) maintain a constant ratio.
Since we have found the ratio of the radii to be \(2\), we can conclude that:
- Circle X can be obtained from Circle Y by scaling up (enlargement) by a factor of \(2\).
Step 3: Conclusion about similarity. Since the ratio of the radii is constant, we conclude that Circle X is similar to Circle Y. Therefore, we have proven that the two circles are similar.
Thus, the circles X and Y are similar.
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