Slope criteria for parallel and perpendicular lines
Slope Criteria for Parallel and Perpendicular Lines
Slope Criteria for Parallel and Perpendicular Lines is a topic in Coordinate Geometry in the Common Core State Standards. It is aligned to Standard G-GPE.5, which requires students to prove the slope criteria for parallel and perpendicular lines and use them to solve problems.
Parallel lines have equal slopes, and perpendicular lines have slopes whose product is \(-1\).
Theory
Two non-vertical lines with slopes \(m_1\) and \(m_2\):
- Parallel \(\iff m_1=m_2\) (same steepness, never meet).
- Perpendicular \(\iff m_1 m_2=-1\), i.e. \(m_2=-\dfrac{1}{m_1}\) (opposite reciprocal).
The slope criteria:
How to classify a pair of lines
- Find each slope with \(m=\dfrac{y_2-y_1}{x_2-x_1}\).
- Equal? The lines are parallel.
- Product \(-1\)? The lines are perpendicular.
- Neither? The lines are neither parallel nor perpendicular.
Compare slopes.
| \(m_1\) | \(=\) | \(\dfrac{2-1}{2-0}=\dfrac12\) |
| \(m_2\) | \(=\) | \(\dfrac{2-0}{5-1}=\dfrac24=\dfrac12\) |
Equal slopes, so the lines are parallel.
Multiply the slopes.
| \(m_1 m_2\) | \(=\) | \(3\cdot\left(-\dfrac13\right)=-1\) |
The product is \(-1\), so the lines are perpendicular.
Take the opposite reciprocal.
| \(m_\perp\) | \(=\) | \(-\dfrac{5}{2}\) |
Equal slopes and the lines are distinct, so they are parallel (product \(16\neq-1\), so not perpendicular).
Common pitfalls
Frequently asked questions
How do you know if two lines are parallel?
They have equal slopes (and different intercepts).
How do you know if two lines are perpendicular?
Their slopes multiply to \(-1\); each is the opposite reciprocal of the other.
What is the opposite reciprocal of a slope?
Flip the fraction and change the sign: \(\dfrac23\) becomes \(-\dfrac32\).
Are horizontal and vertical lines perpendicular?
Yes, but the product rule doesn't apply because a vertical line has no slope.