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Polygon [tex]$ABCD$[/tex] with vertices at [tex]$A(1,-2), B(3,-2), C(3,-4)$[/tex], and [tex]$D(1,-4)$[/tex] is dilated to create polygon [tex]$A'B'C'D'$[/tex] with vertices at [tex]$A'(4,-8), B'(12,-8), C'(12,-16)$[/tex], and [tex]$D'(4,-16)$[/tex].

Determine the scale factor used to create the image.

A. [tex]$\frac{1}{4}$[/tex]
B. [tex]$\frac{1}{2}$[/tex]
C. 2
D. 4



Answer :

GinnyAnswer
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].

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