Equations of parallel and perpendicular lines
Equations of Parallel and Perpendicular Lines
Equations of Parallel and Perpendicular Lines is a topic in Coordinate Geometry in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.2(C), which requires students to determine an equation of a line parallel or perpendicular to a given line through a given point.
To write a parallel or perpendicular line, keep the slope (parallel) or use its opposite reciprocal (perpendicular) in point-slope form.
Theory
To write a line through a point \((x_1,y_1)\), use point-slope form \(y-y_1=m(x-x_1)\). The slope \(m\) comes from the relationship:
- Parallel to a line of slope \(m\): use the same slope \(m\).
- Perpendicular to a line of slope \(m\): use the opposite reciprocal \(-\dfrac1m\).
Point-slope form with the right slope:
How to write the line
- Find the given slope \(m\).
- Choose the new slope: same for parallel, opposite reciprocal for perpendicular.
- Substitute the point and slope into \(y-y_1=m(x-x_1)\).
- Simplify to slope-intercept form if required.
Parallel means the same slope \(m=\dfrac12\); use point-slope.
| \(y-5\) | \(=\) | \(\dfrac12(x-2)\) |
| \(y\) | \(=\) | \(\dfrac12 x-1+5\) |
| \(y\) | \(=\) | \(\dfrac12 x+4\) |
Perpendicular slope is the opposite reciprocal of \(2\), namely \(-\dfrac12\).
| \(y-1\) | \(=\) | \(-\dfrac12(x-4)\) |
| \(y\) | \(=\) | \(-\dfrac12 x+2+1\) |
| \(y\) | \(=\) | \(-\dfrac12 x+3\) |
Slope of the given line is \(\dfrac{8-2}{3-1}=3\); use it.
| \(y-0\) | \(=\) | \(3(x-0)\) |
| \(y\) | \(=\) | \(3x\) |
\(y=7\) is horizontal, so the perpendicular is vertical through \(x=5\).
| \(x\) | \(=\) | \(5\) |
Common pitfalls
Frequently asked questions
How do you write a line parallel to a given line?
Keep the same slope and use point-slope form with the given point.
How do you write a line perpendicular to a given line?
Use the opposite-reciprocal slope and point-slope form with the given point.
What is point-slope form?
\(y-y_1=m(x-x_1)\), a line of slope \(m\) through \((x_1,y_1)\).
What line is perpendicular to a horizontal line?
A vertical line \(x=\text{constant}\), and a horizontal line is perpendicular to a vertical one.