Parallel & Perpendicular Lines
Purplemath
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 have to be a decreasing line). So perpendicular lines have slopes which 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. (This is the non-obvious thing about the slopes of perpendicular lines.) Put this together with the sign change, and you get that the slope of a 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.
To give a numerical example of "negative reciprocals", if the one line's slope is , then the perpendicular line's slope will be . Or, if the one line's slope is m = −2, then the perpendicular line's slope will be . (Remember that any integer can be turned into a fraction by putting it over 1.)
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:
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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 two slopes. So I can keep things straight and tell the difference between the two slopes, I'll use subscripts.
Since these two lines have identical slopes, then:
these lines are parallel.
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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?
I'll find the values of the slopes.
If I were to convert the "3" to fractional form by putting it over "1", then flip it and change its sign, I would get " . This negative reciprocal of the first slope matches the value of the second slope. In other words, these slopes are negative reciprocals, so:
the lines are perpendicular.
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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 also not negative reciprocals, so the lines are not perpendicular. Then the answer is:
these lines are neither.
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Find the slope of a line perpendicular to the line y = −4x + 9.
They've given me the original line's equation, and it's in "y=" form, so it's easy to find the slope. I can just read the value off the equation: m = −4.
This slope can be turned into a fraction by putting it over 1, so this slope can be restated as:
To get the negative reciprocal, I need to flip this fraction, and change the sign. Then the slope of any line perpendicular to the given line is:
Warning: When asked a question of this type ("are these lines 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. The only way to be sure of your answer is to do the algebra.
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