To find the scale factor by which the line segment [tex]$\overline{AB}$[/tex] is dilated to form [tex]$\overline{A'B'}$[/tex], follow these steps:
1. Identify the original and new coordinates:
- Original points: [tex]\( A(0, 2) \)[/tex] and [tex]\( B(2, 3) \)[/tex]
- New points after dilation: [tex]\( A'(0, 4) \)[/tex] and [tex]\( B'(4, 6) \)[/tex]
2. Calculate the distance from the origin to each point:
- Distance from origin to [tex]\(A\)[/tex]:
[tex]\[
\sqrt{(0 - 0)^2 + (2 - 0)^2} = \sqrt{0 + 4} = \sqrt{4} = 2
\][/tex]
- Distance from origin to [tex]\(B\)[/tex]:
[tex]\[
\sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{4 + 9} = \sqrt{13}
\][/tex]
- Distance from origin to [tex]\(A'\)[/tex]:
[tex]\[
\sqrt{(0 - 0)^2 + (4 - 0)^2} = \sqrt{0 + 16} = \sqrt{16} = 4
\][/tex]
- Distance from origin to [tex]\(B'\)[/tex]:
[tex]\[
\sqrt{(4 - 0)^2 + (6 - 0)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}
\][/tex]
3. Determine the scale factor:
- Scale factor based on [tex]\(A\)[/tex] and [tex]\(A'\)[/tex]:
[tex]\[
\text{Scale factor} = \frac{\text{Distance to } A'}{\text{Distance to } A} = \frac{4}{2} = 2
\][/tex]
- Scale factor based on [tex]\(B\)[/tex] and [tex]\(B'\)[/tex]:
[tex]\[
\text{Scale factor} = \frac{\text{Distance to } B'}{\text{Distance to } B} = \frac{2\sqrt{13}}{\sqrt{13}} = 2
\][/tex]
Since both calculations yield the same result, the scale factor by which [tex]$\overline{AB}$[/tex] is dilated to form [tex]$\overline{A'B'}$[/tex] is:
[tex]\[
\boxed{2}
\][/tex]
