To prove that Circle A and Circle C are similar, we need to show that they have the same shape and their respective sizes can be related through a constant scale factor.
Step 1: Identify the properties of the circles.
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Circle A:
- Center: \( (3, 4) \)
- Radius: \( 5 \)
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Circle C:
- Center: \( (0, -1) \)
- Radius: \( 2 \)
Step 2: Understand the concept of similarity in circles.
Circles are considered similar if:
- They have the same shape.
- The ratio of their corresponding radii is constant.
Step 3: Compute the ratio of the radii of the circles.
The radius of Circle A is \( 5 \) and the radius of Circle C is \( 2 \). The ratio of the radii is given by:
\[ \text{Ratio} = \frac{\text{Radius of Circle A}}{\text{Radius of Circle C}} = \frac{5}{2} \]
Step 4: Confirm that the circles have the same shape.
Both circles have the same geometric properties; they are both circles. The definition of a circle does not depend on position in the Cartesian plane or size, but solely on the location of the center and the radius.
Since Circle A and Circle C are both circles, they share the same characteristics as geometric objects (they are both closed shapes where all points are equidistant from a center point).
Step 5: Conclusion.
Because both circles have the same shape (they are both perfect circles) and because their radii have a constant ratio of \( \frac{5}{2} \), we conclude that Circle A and Circle C are similar.
Thus, Circle A and Circle C are similar circles.