To determine the length of [tex]\(\overline{A'B'}\)[/tex] after dilation, we need to follow these steps:
1. Calculate the length of [tex]\(\overline{AB}\)[/tex] using the distance formula:
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here, the coordinates of the endpoints of [tex]\(\overline{AB}\)[/tex] are [tex]\(A(0, -7)\)[/tex] and [tex]\(B(8, 8)\)[/tex]. Plugging these values into the distance formula:
[tex]\[
\text{Length of } \overline{AB} = \sqrt{(8 - 0)^2 + (8 - (-7))^2}
\][/tex]
Simplify within the square root:
[tex]\[
\text{Length of } \overline{AB} = \sqrt{8^2 + (8 + 7)^2}
\][/tex]
[tex]\[
\text{Length of } \overline{AB} = \sqrt{64 + 225}
\][/tex]
[tex]\[
\text{Length of } \overline{AB} = \sqrt{289}
\][/tex]
[tex]\[
\text{Length of } \overline{AB} = 17
\][/tex]
2. Dilation by a scale factor of 2:
Dilation involves scaling a figure larger or smaller by multiplying the distance from the center of dilation (in this case, the origin) by the scale factor. The scale factor of the dilation is given as 2.
Therefore, the length of [tex]\(\overline{A'B'}\)[/tex] after dilation will be twice the original length of [tex]\(\overline{AB}\)[/tex]:
[tex]\[
\text{Length of } \overline{A'B'} = 2 \times \text{Length of } \overline{AB}
\][/tex]
Substitute the length of [tex]\(\overline{AB}\)[/tex]:
[tex]\[
\text{Length of } \overline{A'B'} = 2 \times 17
\][/tex]
[tex]\[
\text{Length of } \overline{A'B'} = 34
\][/tex]
So, the length of [tex]\(\overline{A'B'}\)[/tex] after dilation is [tex]\(\boxed{34}\)[/tex].
