To determine the scale factor by which line segment [tex]\(\overline{AB}\)[/tex] was dilated to create [tex]\(\overline{A'B'}\)[/tex], we'll follow these steps:
1. Identify the coordinates of [tex]\(A'\)[/tex] and [tex]\(B'\)[/tex]:
- [tex]\(A'(0, 4)\)[/tex]
- [tex]\(B'(4, 6)\)[/tex]
2. Calculate the distance from the origin (0,0) to [tex]\(A'\)[/tex]:
[tex]\[ \text{Distance from origin to } A' = \sqrt{(0-0)^2 + (4-0)^2} = \sqrt{0 + 16} = \sqrt{16} = 4 \][/tex]
3. Calculate the distance from the origin (0,0) to [tex]\(B'\)[/tex]:
[tex]\[ \text{Distance from origin to } B' = \sqrt{(4-0)^2 + (6-0)^2} = \sqrt{16 + 36} = \sqrt{52} \approx 7.211 \][/tex]
4. Determine the scale factor:
- The scale factor [tex]\(k\)[/tex] can be found by dividing the distance from the origin to [tex]\(B'\)[/tex] by the distance from the origin to [tex]\(A'\)[/tex]:
[tex]\[ k = \frac{\text{Distance from origin to } B'}{\text{Distance from origin to } A'} = \frac{7.211}{4} \approx 1.803 \][/tex]
Thus, the scale factor [tex]\(k\)[/tex] by which [tex]\(\overline{AB}\)[/tex] was dilated to create [tex]\(\overline{A'B'}\)[/tex] is approximately 1.803. Therefore, the correct scale factor closest to this value in the given options is:
[tex]\[ \boxed{2} \][/tex]
