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To determine which condition confirms that two line segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are perpendicular to each other, we need to recall a fundamental property of perpendicular lines in a coordinate system.
Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex].
Consider the slopes of the lines [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex]:
1. The slope of line segment [tex]\(\overline{AB}\)[/tex] can be calculated as:
[tex]\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
2. The slope of line segment [tex]\(\overline{CD}\)[/tex] can be calculated as:
[tex]\[ m_{CD} = \frac{y_4 - y_3}{x_4 - x_3} \][/tex]
For the line segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] to be perpendicular, the product of their slopes must equal [tex]\(-1\)[/tex]:
[tex]\[ m_{AB} \times m_{CD} = -1 \][/tex]
Substituting the expressions for the slopes:
[tex]\[ \frac{y_2 - y_1}{x_2 - x_1} \times \frac{y_4 - y_3}{x_4 - x_3} = -1 \][/tex]
Now, we need to identify the correct option from the given choices that matches this condition.
Looking at the options provided:
A. [tex]\(\frac{y_4 - y_2}{x_4 - x_2} \times \frac{y_3 - y_1}{x_3 - x_1} = 1\)[/tex]
- This option does not match our required condition.
B. [tex]\(\frac{y_4 - y_3}{y_2 - x_1} + \frac{x_4 - x_3}{x_2 - x_1} = 0\)[/tex]
- This representation is incorrect and mixes variables in an inconsistent manner.
C. [tex]\(\frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_2 - y_1}{x_2 - x_1} = -1\)[/tex]
- This option matches our derived condition exactly.
D. [tex]\(\frac{y_2 - y_1}{x_4 - x_3} - \frac{x_2 - x_1}{y_4 - y_3} = 1\)[/tex]
- This option does not match the necessary form of having a product of slopes.
E. [tex]\(\frac{y_4 - y_2}{y_2 - x_1} + \frac{x_4 - x_2}{x_2 - x_1} = 0\)[/tex]
- This representation is incorrect and mixes variables in an inconsistent manner.
Thus, the correct condition for proving that the two line segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are perpendicular is given by:
[tex]\[ \boxed{\text{C}} \][/tex]
The correct answer is option C.
Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex].
Consider the slopes of the lines [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex]:
1. The slope of line segment [tex]\(\overline{AB}\)[/tex] can be calculated as:
[tex]\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
2. The slope of line segment [tex]\(\overline{CD}\)[/tex] can be calculated as:
[tex]\[ m_{CD} = \frac{y_4 - y_3}{x_4 - x_3} \][/tex]
For the line segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] to be perpendicular, the product of their slopes must equal [tex]\(-1\)[/tex]:
[tex]\[ m_{AB} \times m_{CD} = -1 \][/tex]
Substituting the expressions for the slopes:
[tex]\[ \frac{y_2 - y_1}{x_2 - x_1} \times \frac{y_4 - y_3}{x_4 - x_3} = -1 \][/tex]
Now, we need to identify the correct option from the given choices that matches this condition.
Looking at the options provided:
A. [tex]\(\frac{y_4 - y_2}{x_4 - x_2} \times \frac{y_3 - y_1}{x_3 - x_1} = 1\)[/tex]
- This option does not match our required condition.
B. [tex]\(\frac{y_4 - y_3}{y_2 - x_1} + \frac{x_4 - x_3}{x_2 - x_1} = 0\)[/tex]
- This representation is incorrect and mixes variables in an inconsistent manner.
C. [tex]\(\frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_2 - y_1}{x_2 - x_1} = -1\)[/tex]
- This option matches our derived condition exactly.
D. [tex]\(\frac{y_2 - y_1}{x_4 - x_3} - \frac{x_2 - x_1}{y_4 - y_3} = 1\)[/tex]
- This option does not match the necessary form of having a product of slopes.
E. [tex]\(\frac{y_4 - y_2}{y_2 - x_1} + \frac{x_4 - x_2}{x_2 - x_1} = 0\)[/tex]
- This representation is incorrect and mixes variables in an inconsistent manner.
Thus, the correct condition for proving that the two line segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are perpendicular is given by:
[tex]\[ \boxed{\text{C}} \][/tex]
The correct answer is option C.
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