To solve this problem, we need to determine the length of segment [tex]\( \overline{A'B'} \)[/tex] after dilation when the original segment [tex]\( \overline{AB} \)[/tex] is dilated by a scale factor of 2.
First, we start by identifying the coordinates of the endpoints of [tex]\( \overline{AB} \)[/tex]:
- Coordinates of point [tex]\( A \)[/tex]: [tex]\( (0, -7) \)[/tex]
- Coordinates of point [tex]\( B \)[/tex]: [tex]\( (8, 8) \)[/tex]
Next, we need to calculate the length of segment [tex]\( \overline{AB} \)[/tex] using the distance formula:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Plugging in the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
d_{AB} = \sqrt{(8 - 0)^2 + (8 - (-7))^2}
\][/tex]
[tex]\[
d_{AB} = \sqrt{8^2 + (8 + 7)^2}
\][/tex]
[tex]\[
d_{AB} = \sqrt{64 + 15^2}
\][/tex]
[tex]\[
d_{AB} = \sqrt{64 + 225}
\][/tex]
[tex]\[
d_{AB} = \sqrt{289}
\][/tex]
[tex]\[
d_{AB} = 17
\][/tex]
Now, we need to find the length of segment [tex]\( \overline{A'B'} \)[/tex] after the dilation by a scale factor of 2. In a dilation with the center at the origin, each length is scaled by the scale factor. Therefore, the length of [tex]\( \overline{A'B'} \)[/tex] is:
[tex]\[
d_{A'B'} = 2 \times d_{AB}
\][/tex]
[tex]\[
d_{A'B'} = 2 \times 17
\][/tex]
[tex]\[
d_{A'B'} = 34
\][/tex]
Therefore, the length of [tex]\( \overline{A'B'} \)[/tex] is [tex]\( 34 \)[/tex].
