Math101 · Independent Ontario learning library
Math101learn.math101.caPerpendicular Lines
Perpendicular lines meet at a right angle; their nonzero finite slopes are negative reciprocals.
Perpendicular lines intersect at $90^\circ$. For nonvertical lines with nonzero slopes, $m_1m_2=-1$.
Negative reciprocal slopes
If one slope is $m$, a perpendicular slope is $-1/m$. Reverse numerator and denominator, then change the sign.
A line with slope $2/5$ is perpendicular to one with slope $-5/2$. Their product is $-1$.
Why the rule works
Direction vectors for slopes $m_1$ and $m_2$ can be written $(1,m_1)$ and $(1,m_2)$. Perpendicular vectors have dot product zero:
so $m_1m_2=-1$. This connects the slope rule to vector geometry.
Writing a perpendicular line
Substitute the point and multiply slopes to verify both requirements.
Horizontal and vertical lines
The negative-reciprocal calculation excludes slope $0$ because dividing by zero is undefined. Geometrically, every horizontal line $y=b$ is perpendicular to every vertical line $x=a$.
Recognize this special pair rather than saying their slope product equals $-1$.
Testing from coordinates
To test whether segments $AB$ and $CD$ are perpendicular, calculate their slopes. If one is horizontal and the other vertical, they are perpendicular. Otherwise check whether slopes multiply to $-1$.
For shared-vertex segments, a Pythagorean distance check can also verify a right angle.
Perpendicular bisectors
A perpendicular bisector passes through a segment’s midpoint and meets it at $90^\circ$. To write one:
- find the segment slope;
- take its perpendicular slope;
- find the midpoint;
- use point-slope form.
Every point on this line is equally distant from the segment’s endpoints.
Distance to a line
The shortest distance from a point to a line follows a perpendicular segment. For line $Ax+By+C=0$ and point $(x_0,y_0)$,
This is an extension of the same right-angle geometry.
Applications
Perpendicularity appears in construction, coordinate proofs, tangent-radius relationships in circles, normal lines to curves, navigation, and computer graphics. A right-angle marker in a diagram is evidence; a picture that merely looks square is not.
Common mistakes
Changing only the sign. The slope perpendicular to $2/3$ is $-3/2$, not $-2/3$.
Taking a negative reciprocal for parallel lines. Parallel slopes are equal.
Applying $m_1m_2=-1$ to vertical lines. Use the horizontal–vertical special case.
Finding the correct slope but ignoring the required point. Both conditions define the line.
Quick self-check
- Did I reverse the fraction and change its sign?
- Is there a horizontal–vertical special case?
- Does the new equation pass through the given point?
- Can a slope product or geometric check verify $90^\circ$?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What slope is perpendicular to a line with slope 2/3?
- The reciprocal of 2/3 is 3/2.
- Change the sign.
- The perpendicular slope is −3/2.
End of lesson
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