To determine how the line segments [tex]\(\overline{A B}\)[/tex] and [tex]\(\overline{C D}\)[/tex] are related, we evaluate their slopes and compare them. We have the coordinates of the endpoints as follows:
- [tex]\( A(3, 6) \)[/tex]
- [tex]\( B(8, 7) \)[/tex]
- [tex]\( C(3, 3) \)[/tex]
- [tex]\( D(8, 4) \)[/tex]
### Step-by-Step Solution:
#### Step 1: Calculate the slope of [tex]\(\overline{A B}\)[/tex]
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For [tex]\(\overline{A B}\)[/tex], using points [tex]\(A(3, 6)\)[/tex] and [tex]\(B(8, 7)\)[/tex]:
[tex]\[
m_{AB} = \frac{7 - 6}{8 - 3} = \frac{1}{5} = 0.2
\][/tex]
#### Step 2: Calculate the slope of [tex]\(\overline{C D}\)[/tex]
Using the same formula for the slope, for points [tex]\(C(3, 3)\)[/tex] and [tex]\(D(8, 4)\)[/tex]:
[tex]\[
m_{CD} = \frac{4 - 3}{8 - 3} = \frac{1}{5} = 0.2
\][/tex]
#### Step 3: Compare the slopes
The slopes for both line segments are:
[tex]\[
m_{AB} = 0.2
\][/tex]
[tex]\[
m_{CD} = 0.2
\][/tex]
Since the slopes [tex]\(m_{AB}\)[/tex] and [tex]\(m_{CD}\)[/tex] are equal, this indicates that [tex]\(\overline{A B}\)[/tex] is parallel to [tex]\(\overline{C D}\)[/tex].
### Conclusion:
Given that the slopes are equal, the correct statement describing the relationship between [tex]\(\overline{A B}\)[/tex] and [tex]\(\overline{C D}\)[/tex] is:
A. [tex]\( \overline{A B} \parallel \overline{C D} \)[/tex]
Thus, the line segments [tex]\(\overline{A B}\)[/tex] and [tex]\(\overline{C D}\)[/tex] are parallel.
