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The coordinates of the endpoints of [tex]\overline{AB}[/tex] and [tex]\overline{CD}[/tex] are [tex]A(3,6)[/tex], [tex]B(8,7)[/tex], [tex]C(3,3)[/tex], and [tex]D(8,4)[/tex]. Which statement describes how [tex]\overline{AB}[/tex] and [tex]\overline{CD}[/tex] are related?

A. [tex]\overline{AB} \parallel \overline{CD}[/tex]

B. [tex]\overline{AB} \perp \overline{CD}[/tex], and [tex]\overline{AB}[/tex] bisects [tex]\overline{CD}[/tex].

C. [tex]\overline{AB} \perp \overline{CD}[/tex], but [tex]\overline{AB}[/tex] does not bisect [tex]\overline{CD}[/tex].

D. [tex]\overline{AB}[/tex] is neither parallel nor perpendicular to [tex]\overline{CD}[/tex].



Answer :

GinnyAnswer
To determine the relationship between the lines \( \overline{AB} \) and \( \overline{CD} \), let's analyze their slopes and other properties in a step-by-step manner:

1. Calculate the slope of \( \overline{AB} \):
The slope \( m_{AB} \) is calculated using the formula:
[tex]\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here, \( A = (3, 6) \) and \( B = (8, 7) \).

So,
[tex]\[ m_{AB} = \frac{7 - 6}{8 - 3} = \frac{1}{5} \][/tex]

2. Calculate the slope of \( \overline{CD} \):
The slope \( m_{CD} \) is calculated using the same slope formula:
[tex]\[ m_{CD} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here, \( C = (3, 3) \) and \( D = (8, 4) \).

So,
[tex]\[ m_{CD} = \frac{4 - 3}{8 - 3} = \frac{1}{5} \][/tex]

3. Compare the slopes of \( \overline{AB} \) and \( \overline{CD} \):
Since both slopes are equal, \( m_{AB} = m_{CD} = \frac{1}{5} \), this implies that the lines \( \overline{AB} \) and \( \overline{CD} \) are parallel.

4. Conclusion:
Since the slopes are equal, the correct statement is:
[tex]\[ \overline{A B} \parallel \overline{C D} \][/tex]

Thus, the correct answer is:
A. [tex]\( \overline{A B} \| \overline{C D} \)[/tex]

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