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

Parallel lines and their slopes are easy. Since slope is a measure of the angle of a line from the 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 their values are reciprocals; that is, you take the one slope value, and flip it upside down. Put this together with the sign change, and you get that the slope of the perpendicular line is the "negative reciprocal" of the slope of the original line — and two lines with slopes that are negative reciprocals of each other are perpendicular to each other. In numbers, if the one line's slope is m = 4/5, then the perpendicular line's slope will be m = 5/4. If the one line's slope is m = 2, then the perpendicular line's slope will be m = 1/2.

In your homework, you will probably be given some pairs of points, and be asked to state 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 that 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?

To answer this question, I'll find the slopes.

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?

If I were to flip the "3" and then change its sign, I would get "1/3". In other words, 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?

I'll find the slopes.

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

Warning: When asked a question of this type ("are they parallel or perpendicular?"), do not start drawing pictures. If the lines are close to being parallel or close to being perpendicular (or if you draw the lines messily), you can very-easily get the wrong answer from your picture. Besides, they're not asking if the lines look parallel or perpendicular; they're asking if the lines actually are parallel or perpendicular. To be sure of your answer, do the algebra.