Read this page to see the special relationship of the slopes of parallel and perpendicular lines. After you read, try a few practice problems.
If you know the slopes of two non-vertical lines, can you easily determine if they are parallel? Perpendicular? Yes!
Perpendicular lines have slopes that are opposite reciprocals.
Two lines in a plane are parallel if they never intersect; they have the same slant. We can use slope (or lack of it) to decide if lines are parallel:
Suppose two distinct lines lie in the same coordinate plane.
Then, the two lines are parallel if and only if they are both vertical or they have the same slope.
This statement offers a good opportunity to review the mathematical words if and only if and or.
Now, let's move on to perpendicular lines. Two lines are perpendicular if they intersect at a 90ο angle. For example, the x-axis and y-axis are perpendicular.
Suppose two non-vertical lines lie in the same coordinate plane. Let
The two lines are perpendicular if and only if their slopes are opposite reciprocals:
Equivalently, the two lines are perpendicular if and only if their slopes multiply to
Make sure you understand the opposite reciprocal equation:
|one of the slopes||is||the opposite of||the reciprocal of the other slope|
For example, what is the reciprocal of ? Answer:
What is the opposite [of the] reciprocal of ? Answer:
It is easy to see that this is the correct characterization for perpendicular lines, by studying the sketch below:
The yellow triangle, with base of lengthand right side of length , shows that the slope of the first line is .
Now, imagine that this yellow triangle is a block of wood that is glued to the line. Rotate this block of wood counter-clockwise by 90ο (so the original base is now vertical). Using the rotated triangle to compute the slope of the new, rotated, line gives:
Source: Tree of Math, https://www.onemathematicalcat.org/algebra_book/online_problems/par_perpen.htm
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