Integrated Math 1 / Unit 9: Coordinate Geometry and Constructions
9.3 Parallel and Perpendicular Lines
Notes
Slopes tell you how two lines relate.
- Parallel lines have equal slopes (and different intercepts).
- Perpendicular lines have slopes that are negative reciprocals: $m_1 \cdot m_2 = -1$. Vertical and horizontal lines are also perpendicular.
- To find the perpendicular slope, flip the fraction and change the sign.
- Example: the slope $2$ becomes $\displaystyle -\frac{1}{2}$; the slope $\displaystyle -\frac{3}{4}$ becomes $\displaystyle \frac{4}{3}$.
Practice
What is the slope of a line perpendicular to $y = 2x + 3$?
The negative reciprocal of 2 is $\displaystyle -\frac{1}{2}$.
What is the slope of any line parallel to the line through $(1, 2)$ and $(3, 8)$?
The slope is $\displaystyle \frac{8 - 2}{3 - 1} = 3$, and parallel lines share this slope.
The lines $\displaystyle y = \frac{2}{3}x + 1$ and $3x + 2y = 8$ are:
The second line is $\displaystyle y = -\frac{3}{2}x + 4$, and $\displaystyle \frac{2}{3} \cdot \left(-\frac{3}{2}\right) = -1$.
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