To determine the relationship between line segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex], we'll examine their slopes and any geometric properties related to their intersection.
Step 1: Calculate the slope of [tex]\(\overline{AB}\)[/tex]
The formula to find the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For [tex]\(\overline{AB}\)[/tex]:
- [tex]\(A(3, 6)\)[/tex]
- [tex]\(B(8, 7)\)[/tex]
The slope of [tex]\(\overline{AB}\)[/tex]:
[tex]\[
\text{slope}_{AB} = \frac{7 - 6}{8 - 3} = \frac{1}{5} = 0.2
\][/tex]
Step 2: Calculate the slope of [tex]\(\overline{CD}\)[/tex]
For [tex]\(\overline{CD}\)[/tex]:
- [tex]\(C(3, 3)\)[/tex]
- [tex]\(D(8, 4)\)[/tex]
The slope of [tex]\(\overline{CD}\)[/tex]:
[tex]\[
\text{slope}_{CD} = \frac{4 - 3}{8 - 3} = \frac{1}{5} = 0.2
\][/tex]
Step 3: Compare the slopes
Next, we compare the slopes of [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex]:
- The slope of [tex]\(\overline{AB}\)[/tex] is [tex]\(0.2\)[/tex].
- The slope of [tex]\(\overline{CD}\)[/tex] is [tex]\(0.2\)[/tex].
Since both slopes are equal, we can conclude that [tex]\(\overline{AB}\)[/tex] is parallel to [tex]\(\overline{CD}\)[/tex].
Conclusion:
Given that the slopes are both [tex]\(0.2\)[/tex] and hence equal, the line segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are related as follows:
- Statement A: [tex]\(\overline{AB} \| \overline{CD}\)[/tex]
This means that [tex]\(\overline{AB}\)[/tex] is parallel to [tex]\(\overline{CD}\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{A}
\][/tex]
