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Thursday 18th of April
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Slopes of Perpendicular Lines

Perpendicular lines intersect at a right angle.

If two nonvertical lines are perpendicular, then their slopes are negative reciprocals of each other.

That is, if the slope of one line is , then the slope of any perpendicular line is .

The product of the slopes of two perpendicular lines is -1.

Note: To find the negative reciprocal of a fraction, switch the numerator and the denominator and change the sign. For example, the negative reciprocal of is . The negative reciprocal of -3 is .

Example 1

Find the slope of a line perpendicular to the line that passes through the points (-4, 1) and (1, -2).


First, find the slope of the line through the given points.

Let (x1, y1) = (-4, 1) and (x2, y2) = (1, 2). m
Substitute the values in the slope formula.  

The slope of the line through (-4, 1) and (1, -2) is .

The negative reciprocal of is .

A line perpendicular to the line through (-4, 1) and (1, -2) has slope .

Example 2

Determine if line A is perpendicular to line B.


First, find the slope of each line.


For line A, we use the points (-4, -3) and (2, 4).

The slope of line A is .


Next, find the slope of line B.

We use the points (-3, 0) and (5, -6).

The slope of line B is .  
The slopes, and , are not negative reciprocals so the lines are not perpendicular.

Be careful! The lines do look perpendicular. But the relationship between their slopes tells us the lines are not perpendicular.

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