Related line equations
Check the direction and the supplied point. For distinct nonvertical lines, equal gradients establish parallelism. A product of negative one establishes perpendicularity
- Parallel Keep the gradient, then use the point to find the intercept
- Perpendicular Take the negative reciprocal for oblique lines. Treat horizontal and vertical lines separately
- General form Clear denominators, collect terms on the left, remove a common factor and apply the requested sign convention \(\displaystyle Ax+By+C=0\)
- Coincident lines Equal gradients and equal intercepts describe the same line. Equal gradients with different intercepts describe distinct parallel lines
- Converse checks For nonvertical lines with nonzero gradients, a product of negative one is sufficient to establish perpendicularity. Horizontal and vertical lines form the separate zero/undefined case
Worked example
Find the line perpendicular to g through R.
- \(\displaystyle m_g=-\frac12\)Read the original gradient
- \(\displaystyle m_h=-\frac{1}{-1/2}=2\)Take the negative reciprocal
- \(\displaystyle -1=2(3)+c\)Substitute the point
- \(\displaystyle c=-1-6=-7\)Find the intercept
- \(\displaystyle h:\ y=2x-7\)Write the completed equation
- \(\displaystyle 2(3)-7=-1\)Check that the point lies on the new line
Common mistakes
Difficulty levels
Questions about this topic
Can a perpendicular line have no gradient?+
Yes. A vertical line is perpendicular to a horizontal line. Use its fixed horizontal coordinate rather than taking a reciprocal of zero
Why check the point as well as the gradient?+
Many different lines have the same direction. The point determines which one is required
Are equal gradients enough to identify distinct parallel lines?+
Check the intercepts too. Equal gradients and different intercepts give distinct parallel lines. Equal gradients and equal intercepts give the same line
Why use equal unit lengths when judging a right angle?+
Different horizontal and vertical scale lengths can distort the displayed angle. Use the numerical gradient relationship; an equal-unit diagram shows the geometry without that distortion
Is primitive general form a new kind of line?+
No. It is the same line written with integer coefficients, no common factor and the first nonzero coefficient positive. Follow this convention when the question requests it
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 6: Linear graphs