Question

circle c was dilated to form circle c.
part a
what point is the center of the dilation?
a. (-2, -2) b. (-2, 2)
c. (0, 0) d. (2, -2)
part b
what is the scale factor of the dilation? enter the answer in the box.

Explanation:

Part A

To find the center of dilation, we look for the point that is the same for both circles (the center of both the original and dilated circle). From the graph and the options, the center of dilation should be the center of both circles. The center of circle \( C \) and \( C' \) is at \( (-2, 2) \) as it's the common center for the concentric - like circles (after dilation, the center remains the same for a dilation of a circle). So the correct option is B. \( (-2, 2) \)

Step 1: Determine radii of circles

Let's assume the radius of the smaller circle (original, circle \( C \)) and the larger circle (dilated, circle \( C' \)). From the graph, if we consider the distance from the center \( (-2,2) \) to the edge of the smaller circle and the larger circle. Let's say the radius of the smaller circle \( r = 2 \) units (approximate from the grid) and the radius of the larger circle \( R=4 \) units (approximate from the grid).

Step 2: Calculate scale factor

The scale factor \( k \) of a dilation for a circle is given by the ratio of the radius of the dilated circle to the radius of the original circle. So \( k=\frac{R}{r} \). If \( r = 2 \) and \( R = 4 \), then \( k=\frac{4}{2}=2 \)? Wait, maybe my initial assumption of radii is wrong. Wait, maybe the smaller circle has radius \( r = 1 \) and the larger has \( r = 2 \)? No, looking at the grid, the center is at \( (-2,2) \). The smaller circle (inner) seems to have a radius such that from \( x=-2 \), it goes to \( x = 0 \) (distance 2) and the larger circle goes to \( x = 2 \) (distance 4)? Wait, no, maybe the radius of the original circle (smaller) is 2 units (from center \( (-2,2) \) to \( x = 0 \) is 2 units) and the dilated circle (larger) has a radius of 4 units (from center \( (-2,2) \) to \( x = 2 \) is 4 units). Then scale factor \( k=\frac{4}{2} = 2 \)? But the box has 1, maybe I made a mistake. Wait, maybe the two circles have the same center, and if the original circle and the dilated circle have radii such that the scale factor is 2? Wait, no, maybe the radius of the smaller circle is 2 and the larger is 4, so scale factor is \( \frac{4}{2}=2 \)? But the given box has 1, maybe the user made a typo or my analysis is wrong. Wait, maybe the circles are such that the radius of the original is 2 and the dilated is also 2? No, the circles are different sizes. Wait, perhaps the center is \( (-2,2) \), and the radius of the smaller circle (circle \( C \)) is 2 (distance from \( (-2,2) \) to \( (0,2) \) is 2) and the radius of the larger circle (circle \( C' \)) is 4 (distance from \( (-2,2) \) to \( (2,2) \) is 4). So scale factor \( k=\frac{4}{2}=2 \). But the initial box has 1, maybe that's incorrect. Wait, maybe the problem is that the two circles have the same center, and if we consider the distance from the center to a point on the circle. Let's take a point on the smaller circle: say \( (0,2) \) (center \( (-2,2) \), so distance is 2). A point on the larger circle: \( (2,2) \), distance from center \( (-2,2) \) is 4. So scale factor is \( \frac{4}{2}=2 \).

Answer:

B. \( (-2, 2) \)

Part B