Perpendicular Lines and Equations

Perpendicular lines intersect to form a $90^\circ$ angle.
The slopes of perpendicular lines are negative reciprocals of each other (negated and inverted).
The product of two perpendicular slopes is always $-1$ : $\frac{3}{2} \times \left(-\frac{2}{3}\right) = -\frac{6}{6} = -1$
Identifying Perpendicular Lines:
- $y = -2x + 7$ and $y = \frac{1}{2}x - 94$ are perpendicular because $-2$ and $\frac{1}{2}$ are negative reciprocals.
- $y = \frac{3}{4}x - 79$ and $y = -\frac{3}{4}x + 0.5$ are not perpendicular because their slopes are negated but not inverted.

To find the equation of a line perpendicular to a given line passing through a specific point $(x_1, y_1)$ :
1. Determine the perpendicular slope ( $m$ ) by taking the negative reciprocal of the original slope.
2. Substitute $m$ and $(x_1, y_1)$ into point-slope form: $y - y_1 = m(x - x_1)$
3. Simplify the equation to slope-intercept form ( $y = mx + b$ ).
Example 1: Line perpendicular to $y = -4x + \frac{2}{3}$ passing through $(8, 5)$ :
- Original slope: $-4$
- Perpendicular slope: $\frac{1}{4}$
- Point-slope form: $y - 5 = \frac{1}{4}(x - 8)$
- Distributed form: $y - 5 = \frac{1}{4}x - 2$
- Final equation: $y = \frac{1}{4}x + 3$
Example 2: Line perpendicular to $y = -\frac{2}{5}x - 4$ passing through $(-3, 1)$ :
- Original slope: $-\frac{2}{5}$
- Perpendicular slope: $\frac{5}{2}$
- Point-slope form: $y - 1 = \frac{5}{2}(x - (-3))$
- Distributed form: $y - 1 = \frac{5}{2}x + \frac{15}{2}$
- Final equation: $y = \frac{5}{2}x + \frac{17}{2}$