Determine if lines PQ and RS are parallel, perpendicular, or neither, given P(-3, 14), Q(2, -1), R(4, 8), S(-2, -10).
Understand the Problem
The question asks us to determine if lines PQ and RS are parallel, perpendicular, or neither. To do this, calculate the slopes of both lines and then compare them. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular. If neither of these conditions is met, they are neither parallel nor perpendicular.
Answer
Slope of $\overrightarrow{PQ} = -3$ Slope of $\overrightarrow{RS} = 3$ The lines are neither.
Answer for screen readers
Slope of $\overrightarrow{PQ} = -3$ Slope of $\overrightarrow{RS} = 3$ The lines are neither.
Steps to Solve
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Calculate the slope of line PQ
The slope of a line given two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$. For line PQ, the points are P(-3, 14) and Q(2, -1). $m_{PQ} = \frac{-1 - 14}{2 - (-3)} = \frac{-15}{2 + 3} = \frac{-15}{5} = -3$
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Calculate the slope of line RS
For line RS, the points are R(4, 8) and S(-2, -10). $m_{RS} = \frac{-10 - 8}{-2 - 4} = \frac{-18}{-6} = 3$
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Determine if the lines are parallel, perpendicular, or neither
Parallel lines have equal slopes: $m_1 = m_2$. Perpendicular lines have slopes that are negative reciprocals of each other: $m_1 = -\frac{1}{m_2}$. In this case, $m_{PQ} = -3$ and $m_{RS} = 3$. Since $-3 \neq 3$, the lines are not parallel. To check for perpendicularity: $-\frac{1}{m_{RS}} = -\frac{1}{3} \neq -3$, so the lines are not perpendicular.
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Final Answer The lines are neither parallel nor perpendicular.
Slope of $\overrightarrow{PQ} = -3$ Slope of $\overrightarrow{RS} = 3$ The lines are neither.
More Information
The concept of slope is fundamental in coordinate geometry, describing the steepness and direction of a line. Parallel lines, sharing the same slope, never intersect, while perpendicular lines, with slopes that are negative reciprocals, intersect at a right angle (90 degrees)
Tips
A common mistake is to incorrectly calculate the slope by subtracting the x-coordinates from the y-coordinates or mixing up the order of subtraction. Another common mistake is to incorrectly identify the relationship between the slopes of perpendicular lines, sometimes forgetting the negative sign or not taking the reciprocal.
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