Certainly! To determine the length of \( \overline{A'B'} \), we first need to calculate the original length of \( \overline{AB} \). Here are the steps:
1. Identify the coordinates of points \(A\) and \(B\):
- Point \(A\) has coordinates \((0, -7)\).
- Point \(B\) has coordinates \((8, 8)\).
2. Apply the distance formula:
The distance formula is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
3. Substitute the coordinates of \(A\) and \(B\) into the distance formula:
Here, \((x_1, y_1) = (0, -7)\) and \((x_2, y_2) = (8, 8)\).
[tex]\[
d = \sqrt{(8 - 0)^2 + (8 - (-7))^2}
\][/tex]
4. Simplify the expression:
[tex]\[
d = \sqrt{(8)^2 + (8 + 7)^2}
\][/tex]
[tex]\[
d = \sqrt{64 + 225}
\][/tex]
[tex]\[
d = \sqrt{289}
\][/tex]
[tex]\[
d = 17
\][/tex]
So, the length of \( \overline{AB} \) is \( 17 \).
5. Apply the scale factor of 2:
Since the polygon is dilated by a scale factor of 2, the length of each side of the polygon will also be scaled by a factor of 2.
Therefore, the length of \( \overline{A'B'} \) will be:
[tex]\[
2 \times 17 = 34
\][/tex]
So, the length of [tex]\( \overline{A'B'} \)[/tex] after the dilation is [tex]\( 34 \)[/tex].
