Question

understanding dilations and similarity continued
5 figure ( a ^ { prime } b ^ { prime } c ^ { prime } d ^ { prime } e ^ { prime } f ^ { prime } ) is a dilation of figure ( a b c d e f ).
a. label the center of dilation and explain how you found it.
b. what is the scale factor of the dilation?

Explanation:

Step1: Understand the concept of dilation center

The center of dilation is the fixed point from which all points are expanded or contracted. To find it, we can use the property that the lines connecting corresponding points (e.g., \(A\) and \(A'\), \(B\) and \(B'\) etc.) intersect at the center of dilation.

Step2: Find the scale factor

The scale factor \(k\) of a dilation is given by the ratio of the length of a side of the dilated figure to the length of the corresponding side of the original figure. Let's assume we take a horizontal side. For example, if the length of \(AB\) (original) is \(3\) units and the length of \(A'B'\) (dilated) is \(6\) units.
$$k=\frac{\text{length of dilated side}}{\text{length of original side}}$$

Answer:

a. To label the center of dilation: Draw lines connecting \(A\) to \(A'\), \(B\) to \(B'\), \(C\) to \(C'\), \(D\) to \(D'\), \(E\) to \(E'\), \(F\) to \(F'\). The intersection point of these lines is the center of dilation.
b. If, for example, a side of length \(l\) in \(ABCDEF\) has a corresponding side of length \(2l\) in \(A'B'C'D'E'F'\) (by measuring on the grid), the scale factor is \(2\).