Question

what are the scale factor and center of the dilation? simplify your answers and write them as fractions or whole numbers. scale factor: center of the dilation: ( , )

Explanation:

Step1: Find distance between points

Let's assume a point on the small - circle, say \(L(-2,3)\) and its corresponding point on the large - circle \(L'(4, - 3)\). The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). But we can also use the ratio of the lengths of corresponding line - segments in the coordinate plane.
We can see that the change in \(x\) from \(L\) to \(L'\) is \(4-(-2)=6\) and the change in \(y\) is \(-3 - 3=-6\).

Step2: Determine scale factor

To find the scale factor \(k\), we can compare the lengths of corresponding radii. Let's consider the horizontal distance from the center of dilation (which we will find next) to a point on the circle. If we assume the center of dilation is the origin \((0,0)\) (by observing the symmetry of the dilation), the distance from the origin to \(L(-2,3)\) is \(d_1=\sqrt{(-2 - 0)^2+(3 - 0)^2}=\sqrt{4 + 9}=\sqrt{13}\), and the distance from the origin to \(L'(4,-3)\) is \(d_2=\sqrt{(4 - 0)^2+(-3 - 0)^2}=\sqrt{16 + 9}=\sqrt{25}=5\). Another way is to look at the ratio of the distances of corresponding points from the center of dilation. If we consider the horizontal or vertical displacements, the distance from the center of dilation to a point on the small - circle and its corresponding point on the large - circle. The distance from the center of dilation (origin) to \(L\) in the \(x\) - direction is \(2\) and to \(L'\) is \(4\). The scale factor \(k = 2\) (since \(\frac{4}{2}=2\)).

Step3: Find center of dilation

By observing the transformation of the circles, we can see that the center of dilation is the origin \((0,0)\) because the transformation is symmetric about the origin and the lines connecting corresponding points on the two circles pass through the origin.

Answer:

scale factor: \(2\)
center of the dilation: \((0,0)\)