To determine the scale factor used to create the image of polygon [tex]\(A B C D\)[/tex] from its original vertices to its dilated vertices, we compare the coordinates of corresponding points.
Let's start by comparing the original vertex [tex]\(A\)[/tex] and its dilated image [tex]\(A'\)[/tex].
Original coordinates of [tex]\(A\)[/tex]:
[tex]\[ A(1, -2) \][/tex]
Dilated coordinates of [tex]\(A'\)[/tex]:
[tex]\[ A'(4, -8) \][/tex]
We can find the scale factor, [tex]\(k\)[/tex], by comparing the coordinates of [tex]\(A\)[/tex] and [tex]\(A'\)[/tex]:
1. Calculate the scale factor for the [tex]\(x\)[/tex]-coordinates:
[tex]\[
k_x = \frac{x_{A'}}{x_A} = \frac{4}{1} = 4
\][/tex]
2. Calculate the scale factor for the [tex]\(y\)[/tex]-coordinates:
[tex]\[
k_y = \frac{y_{A'}}{y_A} = \frac{-8}{-2} = 4
\][/tex]
We see that both [tex]\(k_x\)[/tex] and [tex]\(k_y\)[/tex] yield the same result, 4. Therefore, the scale factor for the dilation is [tex]\(4\)[/tex].
Thus, the scale factor used to create the dilated image of polygon [tex]\(A B C D\)[/tex] is [tex]\( \boxed{4} \)[/tex].
