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Given that the coordinates of the points π΄, π΅, πΆ, and π· are negative 15, eight; negative six, 10; negative eight, negative seven; and negative six, negative 16, respectively, determine whether line π΄π΅ and line πΆπ· are parallel, perpendicular, or neither.
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Points π΄ and π΅ fall on the line π΄π΅ and points πΆ and π· fall on the line πΆπ·.
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To classify the lines, we have to remember: parallel lines have the same slope and they do not intersect.
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Perpendicular lines have negative reciprocal slopes and intersect at a 90-degree angle.
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And neither are lines that are not parallel or perpendicular, lines that do intersect but do not form a right angle.
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This means to consider whether or not these lines are parallel or perpendicular, we need to know the slopes of these lines.
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In the general form, π¦ equals ππ₯ plus π, the π represents the slope.
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And we can find the slope π if we have two points by saying π equals π¦ two minus π¦ one over π₯ two minus π₯ one.
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In order to categorize these lines, we need to find the slopes of line π΄π΅ and line πΆπ·.
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We can start with line π΄π΅.
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Let point π΄ be π₯ one, π¦ one and point π΅ be π₯ two, π¦ two.
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Then, the slope will be 10 minus eight over negative six minus negative 15.
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10 minus eight is two.
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Negative six minus negative 15 is negative six plus 15, which is positive nine.
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So, we can say that the slope of line π΄π΅ is two-ninths.
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We repeat this process for line πΆπ·.
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Let πΆ be π₯ one, π¦ one and π· be π₯ two, π¦ two.
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And weβll get π equals negative 16 minus negative seven over negative six minus negative eight.
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Negative 16 minus negative seven is negative 16 plus seven, which is negative nine.
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Negative six minus negative eight is negative six plus eight which is two.
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The slope of line πΆπ· is then negative nine over two.
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If we compared these two slopes, negative nine over two is the negative reciprocal of two over nine.
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And if you werenβt sure, you can multiply them together.
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Reciprocals multiply together to equal one and negative reciprocals multiply together to equal negative one.
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These two slopes are the negative reciprocals of one another, making these lines perpendicular.