Question

circle c was dilated to form circle c.
part a
what point is the center of the dilation?
(-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

Step1: Recall the property of dilation

Dilation is a transformation that changes the size of a figure but not its shape. The center of dilation is the fixed point about which all points are expanded or contracted. For two circles formed by dilation, the center of dilation is the common center of the two circles.
Looking at the graph, the center of both circle \(C\) and circle \(C'\) is at the point \((- 2,2)\).

Part B

Step1: Calculate the radius of each circle

Let's assume the radius of circle \(C\) (the smaller circle). If we count the units from the center \((-2,2)\) to the circumference of circle \(C\) along the \(y\) - axis (for simplicity), the radius \(r\) of circle \(C\) is \(2\) units (from \(y = 2\) to \(y=4\)).
The radius \(R\) of circle \(C'\) (the larger circle) is \(4\) units (from \(y = 2\) to \(y = 6\)).

Step2: Use the formula for the scale factor of dilation

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{R}{r}\), where \(R\) is the radius of the dilated circle (\(C'\)) and \(r\) is the radius of the original circle (\(C\)).
Substitute \(R = 4\) and \(r=2\) into the formula: \(k=\frac{4}{2}=2\)

Answer:

Part A: B. \((-2,2)\)
Part B: \(2\)