38.6 Perpendicular Line Construction

Perpendicular Line Construction is the step-by-step process of building the equation of a new line that passes through a specified point and forms a right angle with a given reference line, by converting the reference slope into its negative reciprocal and applying that value to the new point.

Reference Slope Extraction

Procedure

The process begins by extracting the slope of the given reference line, using whichever method matches the form in which the reference equation is written.

Example

Given the reference line:

the extracted slope is $m=-\frac{2}{3}$.

Perpendicular Slope Determination

Procedure

The negative reciprocal of the extracted slope is computed by exchanging its numerator and denominator and reversing its sign:

Applied Example

For the extracted slope $-\frac{2}{3}$, the perpendicular slope is determined as $\frac{3}{2}$.

Required Point Placement in Point-Slope Form

Procedure

The coordinates of the required point, written as $(x_1,y_1)$, are placed into the point-slope formula alongside the perpendicular slope determined in the previous step.

Applied Example

For the required point $(-3,4)$ and perpendicular slope $\frac{3}{2}$, these values are prepared for substitution into the point-slope structure.

Perpendicular Point-Slope Equation

Formula

The point-slope form of the new line is written as:

Applied Example

Using the required point and perpendicular slope from above:

Perpendicular Equation Simplification

Procedure

The point-slope equation is expanded and simplified to isolate $y$, converting the result into slope-intercept form.

Applied Example

Expanding the previous result:

which simplifies to:

Required Point Substitution Check

Procedure

The required point is substituted back into the newly constructed equation to confirm that it produces a true numerical statement.

Applied Example

Substituting $(-3,4)$ into $y=\frac{3}{2}x+\frac{17}{2}$ gives $4=-\frac{9}{2}+\frac{17}{2}$, which simplifies to a true statement, confirming the point lies on the new line.

Negative Reciprocal Verification

Final Verification

As a last confirmation, the product of the reference slope and the newly constructed slope is computed to confirm it equals negative one.

Applied Example

confirming the perpendicular relationship holds and the construction is complete and correct.