To determine the scale factor of the dilation between the preimage polygon [tex]\(ABCD\)[/tex] and the image polygon [tex]\(PQRS\)[/tex], we begin by identifying the coordinates of corresponding vertices. The coordinates of the preimage are [tex]\((2, 2)\)[/tex], [tex]\((6, 8)\)[/tex], [tex]\((12, 8)\)[/tex], and [tex]\((16, 2)\)[/tex]. The coordinates of the image are [tex]\((11, 15)\)[/tex], [tex]\((9, 12)\)[/tex], [tex]\((6, 12)\)[/tex], and [tex]\((4, 15)\)[/tex].
The scale factor can be found by comparing the distances between corresponding points in the preimage and the image.
1. Calculate the distance between two corresponding points in the preimage and the image:
We'll use the distance formula:
[tex]\[
\text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
2. Select corresponding points:
Let's use [tex]\(A\)[/tex] from the preimage which is [tex]\((2, 2)\)[/tex] and [tex]\(B\)[/tex] which is [tex]\((6, 8)\)[/tex].
Let's use [tex]\(P\)[/tex] from the image which is [tex]\((11, 15)\)[/tex] and [tex]\(Q\)[/tex] which is [tex]\((9, 12)\)[/tex].
3. Compute the distance [tex]\(AB\)[/tex] in the preimage:
[tex]\[
AB = \sqrt{(6 - 2)^2 + (8 - 2)^2}
\][/tex]
[tex]\[
AB = \sqrt{4^2 + 6^2}
\][/tex]
[tex]\[
AB = \sqrt{16 + 36}
\][/tex]
[tex]\[
AB = \sqrt{52}
\][/tex]
[tex]\[
AB = 7.211102550927978
\][/tex]
4. Compute the distance [tex]\(PQ\)[/tex] in the image:
[tex]\[
PQ = \sqrt{(9 - 11)^2 + (12 - 15)^2}
\][/tex]
[tex]\[
PQ = \sqrt{(-2)^2 + (-3)^2}
\][/tex]
[tex]\[
PQ = \sqrt{4 + 9}
\][/tex]
[tex]\[
PQ = \sqrt{13}
\][/tex]
[tex]\[
PQ = 3.605551275463989
\][/tex]
5. Determine the scale factor:
The scale factor [tex]\(k\)[/tex] is the ratio of the corresponding distances:
[tex]\[
k = \frac{\text{distance in image}}{\text{distance in preimage}}
\][/tex]
[tex]\[
k = \frac{PQ}{AB}
\][/tex]
[tex]\[
k = \frac{3.605551275463989}{7.211102550927978}
\][/tex]
[tex]\[
k = 0.5
\][/tex]
Thus, the scale factor of the dilation is [tex]\(\boxed{0.5}\)[/tex].
