Slope: Parallel and Perpendicular Lines


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Slope: Parallel and Perpendicular Lines (page 2 of 2)

Parallel lines are easy: since slope is a measure of the angle of a line from horizontal, and since parallel lines must have the same angle, then parallel lines have the same slope (and lines with the same slope are parallel).

Perpendicular lines are a bit more complicated. If you visualize a line with positive slope (so it's an increasing line), then the perpendicular line must have negative slope (because it will be a decreasing line). So perpendicular slopes have opposite signs. The other "opposite" thing with perpendicular slopes is that they are reciprocals; that is, you take the one slope, and flip it upside down. Put this together with the sign change, and you get that the perpendicular slope is the "negative reciprocal" of the original slope (and lines with slopes that are negative reciprocals are perpendicular). In numbers, if the one slope is m = ⁴/₅, then the perpendicular slope will be m = ^–5/₄. If the one slope is m = –2, then the perpendicular slope will be m = ¹/₂.

For instance, you will probably be given some pairs of points, and be asked whether the lines through the pairs of points are "parallel, perpendicular, or neither". To answer the question, you'll have to calculate the slopes and compare them. Here's how it works:

One line passes through the points (–1, –2) and (1, 2); another line passes through the points (–2, 0) and (0, 4). Are these lines parallel, perpendicular, or neither?

Find the slopes.

m_1 = 2, m_2 = 2

Since these two lines have identical slopes, then these lines are parallel.

One line passes through the points (0, –4) and (–1, –7); another line passes through the points (3, 0) and (–3, 2). Are these lines parallel, perpendicular, or neither?

m_1 = 3, m_2 = -1/3

If you flip "3" and then change the sign, you get "^–1/₃". These slopes are negative reciprocals, so the lines through the points are perpendicular.

One line passes through the points (–4, 2) and (0, 3); another line passes through the points (–3, –2) and (3, 2). Are these lines parallel, perpendicular, or neither?

Find the slopes.

m_1 = 1/4, m_2 = 2/3

These slopes are not the same, so the lines are not parallel. The slopes are not negative reciprocals either, so the lines are not perpendicular. Then the answer is "neither".

When asked a question of this type ("are they parallel or perpendicular?"), do not start drawing pictures. If the lines are close to parallel or close to perpendicular (or if you draw messily), you can get the wrong answer from a picture. And besides, they're not asking if the lines look parallel or perpendicular; they're asking if the lines are parallel or perpendicular. Do the algebra.

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Cite this article as:

Stapel, Elizabeth. "Slope: Parallel and Perpendicular Lines." Purplemath. Available from
http://www.purplemath.com/modules/slope2.htm. Accessed

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