Write the equation of a perpendicular line

Given a line and a point, the learner determines the equation of the unique line through that point that meets the given line at a right angle, interpreting perpendicularity through slopes: for nonvertical, nonhorizontal lines, the slopes are negative reciprocals. The learner represents the result in point-slope, slope-intercept, or standard form and handles horizontal and vertical lines separately; vector methods, three-dimensional lines, and more advanced generalizations are outside this scope.

Detailed Explanation: Write the equation of a perpendicular line

To write the equation of a line perpendicular to a given line:

Find the slope of the given line.
Find its negative reciprocal:
- Change the sign.
- Flip the fraction.
Use the new slope and the given point in point-slope form:

y-y_1=m(x-x_1)

Example

Find the equation of the line perpendicular to

y=\frac{2}{3}x-5

that passes through $(6,1)$ .

Step 1: Identify the given slope.

The equation is in slope-intercept form, $y=mx+b$ , so the slope is

m=\frac{2}{3}.

Step 2: Find the perpendicular slope.

The negative reciprocal of $\frac{2}{3}$ is

m_{\perp}=-\frac{3}{2}.

Step 3: Use point-slope form.

The line must pass through $(6,1)$ , so substitute $x_1=6$ , $y_1=1$ , and $m=-\frac{3}{2}$ :

y-1=-\frac{3}{2}(x-6).

This is the equation of the perpendicular line in point-slope form.

To write it in slope-intercept form, simplify:

y-1=-\frac{3}{2}x+9

y=-\frac{3}{2}x+10.

Therefore, the perpendicular line is

\boxed{y=-\frac{3}{2}x+10}.

Remember the special cases:

A horizontal line has slope $0$ , so a perpendicular line is vertical: $x=c$ .
A vertical line has an undefined slope, so a perpendicular line is horizontal: $y=c$ .

Learn by doing: Write the equation of a perpendicular line

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Line Segment (Points) - Find Perpendicular Bisector (Formula)

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