Geometry / Unit 3: Parallel and Perpendicular Lines
3.6 Equations of Parallel and Perpendicular Lines
Notes
To write the equation of a line through a point, use the slope and point-slope form $y - y_1 = m(x - x_1)$, or solve for $b$ in $y = mx + b$.
- Parallel: use the same slope.
- Perpendicular: use the opposite reciprocal slope.
- Example: perpendicular to $y = 2x + 1$ through $(4, 3)$. Slope is $\displaystyle -\frac{1}{2}$. $\displaystyle 3 = -\frac{1}{2}(4) + b$, so $b = 5$: $\displaystyle y = -\frac{1}{2}x + 5$.
Practice
Which line is parallel to $y = -2x + 5$ and passes through $(1, 4)$?
$4 = -2(1) + b$, so $b = 6$.
Which line is perpendicular to $y = 3x$ and passes through $(3, 2)$?
Slope $\displaystyle -\frac{1}{3}$: $\displaystyle 2 = -\frac{1}{3}(3) + b$, so $b = 3$.
Line $k$ passes through $(0, 0)$ and $(2, 6)$. Line $j$ passes through $(1, 1)$ and $(4, 0)$. How are the lines related?
Slopes are 3 and $\displaystyle -\frac{1}{3}$, and $\displaystyle 3 \cdot \left(-\frac{1}{3}\right) = -1$.
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