Question

Prove that the two circles shown below are similar. (10 points)

Circle A is shown with a center at 3, 4 and a radius of 5. Circle C is shown with a center of 0, negative 1 and a radius of 2.

Answer 1

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.**

- **Circle A:**
- Center: \( (3, 4) \)
- Radius: \( 5 \)

- **Circle C:**
- Center: \( (0, -1) \)
- Radius: \( 2 \)

**Step 2: Understand the concept of similarity in circles.**

Circles are considered similar if:
1. They have the same shape.
2. 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.