Algebra
Linear functions
Parallel and perpendicular lines
20 practice questions
2 video lessons
Theory + worked examples
Theory
Two lines' slopes reveal their relationship:
- Parallel \(\iff m_1=m_2\) (same slope).
- Perpendicular \(\iff m_1 m_2=-1\), i.e. \(m_2=-\dfrac{1}{m_1}\).
Perpendicular slope is the opposite reciprocal: flip and change the sign.
Parallel lines match; a perpendicular one crosses at a right angle.
The slope relationships.
Slope criteria:
\[\text{parallel: } m_1=m_2,\qquad \text{perpendicular: } m_1 m_2=-1\]
Opposite reciprocal: \(\dfrac23\to-\dfrac32\).
How to use the slope criteria
- Find each line's slope.
- Equal slopes \(\Rightarrow\) parallel.
- Product \(-1\) \(\Rightarrow\) perpendicular.
- For a perpendicular line, use the opposite reciprocal.
Example 1 — Parallel line
Write the line parallel to \(y=2x+1\) through \((0,5)\).
Solution
Same slope \(2\), new intercept.
| \(y\) | \(=\) | \(2x+5\) |
Example 2 — Perpendicular slope
What slope is perpendicular to a line of slope \(3\)?
Solution
Opposite reciprocal.
| \(m_\perp\) | \(=\) | \(-\dfrac13\) |
Example 3 — Are they parallel?
Are \(y=\dfrac12x+1\) and \(y=\dfrac12x-4\) parallel?
Solution
Compare slopes.
| \(\dfrac12\) | \(=\) | \(\dfrac12\ \checkmark\) |
Yes — same slope.
Example 4 — Perpendicular line
Write the line perpendicular to \(y=2x\) through \((0,0)\).
Solution
Perpendicular slope is \(-\dfrac12\).
| \(y\) | \(=\) | \(-\dfrac12x\) |
Common pitfalls
Perpendicular is the opposite reciprocal, not just the negative.
Parallel lines never meet and share a slope.
Change both the sign and the fraction for perpendicular.
Frequently asked questions
When are two lines parallel?
When they have equal slopes.
When are two lines perpendicular?
When their slopes multiply to \(-1\).
What is perpendicular to slope \(4\)?
Slope \(-\dfrac14\).
What is the opposite reciprocal of \(\\dfrac23\)?
\(-\dfrac32\).
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