Parallel And Perpendicular Lines
Working with parallel and perpendicular lines is a foundational skill assumed for Queensland Year 11 Mathematical Methods (QCAA). Two lines are parallel when their gradients are equal, and perpendicular when their gradients multiply to give negative one.
You will state a parallel or perpendicular gradient using the negative reciprocal, find the equation of such a line through a given point, classify two lines, and solve for an unknown parameter.
Every question with a fully worked solution.
- Parallel And Perpendicular Lines - Video - Parallel and Perpendicular Lines Watch
Theory
In Year 11 Mathematical Methods (QCAA), two straight lines are parallel when their gradients are equal, \(m_1=m_2\), and perpendicular when their gradients multiply to \(-1\), \(m_1 m_2=-1\). This page shows how to state a parallel or perpendicular gradient (the negative reciprocal), find the equation of such a line through a point, classify two lines, and solve for an unknown parameter.
Two lines are parallel if they have the same gradient, \(m_1=m_2\). They rise at the same rate and never meet.
Two lines are perpendicular if they meet at a right angle. Their gradients satisfy \(m_1 m_2=-1\), so each is the negative reciprocal of the other: \(m_2=-\dfrac{1}{m_1}\). For example, a gradient of \(\tfrac{1}{2}\) has perpendicular gradient \(-2\).
Parallel and perpendicular gradient conditions:
How to find a parallel or perpendicular line
- Gradient of the given line: read \(m\) from \(y=mx+c\) (rearrange first if needed).
- New gradient: for parallel keep \(m\); for perpendicular take the negative reciprocal \(-\dfrac{1}{m}\).
- Line: substitute the new gradient and the given point into \(y-y_1=m(x-x_1)\) and simplify.
Read the gradient of the given line:
| \(m_1\) | \(=\) | \(3\) |
(a) Parallel — equal gradients:
| \(m_{\parallel}\) | \(=\) | \(m_1\) |
| \(=\) | \(3\) |
(b) Perpendicular — negative reciprocal:
| \(m_{\perp}\) | \(=\) | \(-\dfrac{1}{m_1}\) |
| \(=\) | \(-\dfrac{1}{3}\) |
(a) parallel gradient \(3\); (b) perpendicular gradient \(-\dfrac{1}{3}\).
Parallel means the same gradient:
| \(m\) | \(=\) | \(-1\) |
Substitute \(m=-1\) and \((1,1)\) into \(y-y_1=m(x-x_1)\):
| \(y-1\) | \(=\) | \(-1(x-1)\) |
| \(y-1\) | \(=\) | \(-x+1\) |
| \(y\) | \(=\) | \(-x+1+1\) |
| \(y\) | \(=\) | \(-x+2\) |
Equation: \(y=-x+2\).
Gradient of the given line:
| \(m_1\) | \(=\) | \(\dfrac{1}{2}\) |
Perpendicular gradient — negative reciprocal:
| \(m\) | \(=\) | \(-\dfrac{1}{\,1/2\,}\) |
| \(=\) | \(-2\) |
Substitute \(m=-2\) and \((2,2)\):
| \(y-2\) | \(=\) | \(-2(x-2)\) |
| \(y-2\) | \(=\) | \(-2x+4\) |
| \(y\) | \(=\) | \(-2x+6\) |
Equation: \(y=-2x+6\).
Perpendicular gradients multiply to \(-1\):
| \(k\times\dfrac{1}{4}\) | \(=\) | \(-1\) |
Multiply both sides by \(4\):
| \(k\) | \(=\) | \(-1\times 4\) |
| \(k\) | \(=\) | \(-4\) |
Parameter: \(k=-4\).
Common pitfalls
Frequently asked questions
When are two lines parallel?
When their gradients are equal, \(m_1=m_2\). Parallel lines rise at the same rate and never intersect.
When are two lines perpendicular?
When their gradients multiply to \(-1\), \(m_1 m_2=-1\); each gradient is the negative reciprocal of the other.
What is the perpendicular gradient of a line with gradient 2/3?
The negative reciprocal: flip \(\tfrac{2}{3}\) to \(\tfrac{3}{2}\) and negate, giving \(-\tfrac{3}{2}\).
How do you find the equation of a perpendicular line through a point?
Take the negative reciprocal gradient, then substitute it and the point into \(y-y_1=m(x-x_1)\) and simplify.
Are the axes perpendicular under this rule?
They are perpendicular, but the rule \(m_1 m_2=-1\) does not apply directly: the \(y\)-axis is vertical with an undefined gradient, a special case.