Perpendicular Lines

What Are Perpendicular Lines?

Two lines are perpendicular if they intersect to form a right angle ($90$°). In coordinate geometry, the slopes of two perpendicular lines are related in the following way:

$$m_1 \times m_2 = -1$$

where $m_1$ and $m_2$ are the slopes of the lines.

Equation of a Perpendicular Line

If you know the slope $m_1$ of one line, the slope $m_2$ of a line perpendicular to it is:

$$m_2 = -\frac{1}{m_1}, \quad m_1 \neq 0$$

Special Cases:

• A vertical line ($x = k$) is perpendicular to a horizontal line ($y = k$).
• The slope of a vertical line is undefined, while the slope of a horizontal line is $0$.

Finding a Perpendicular Line

To find the equation of a line perpendicular to a given line and passing through a specific point, use the point-slope form:

$$y - y_1 = m(x - x_1)$$

where mm is the slope of the perpendicular line and $(x_1, y_1)$ is the point through which the line passes.

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Worked Examples

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Summary

• Definition: Perpendicular lines intersect to form a right angle.
• Key Property: The slopes of two perpendicular lines satisfy $m_1 \times m_2 = -1$
• Special Cases: Vertical lines ($x = k$) and horizontal lines ($y = k$) are perpendicular.
• To find the equation of a perpendicular line:
  1. Determine the negative reciprocal of the slope.
  2. Use the point-slope formula.
• Practice identifying perpendicular lines and deriving their equations to strengthen understanding.