Parallel and Perpendicular Lines: Algebraic Tests and Geometric Interpretation

Parallel and perpendicular lines can be recognised geometrically and tested algebraically. Coordinate geometry is strongest when both views agree.

1. Parallel nonvertical lines have equal gradients

If y=m₁x+c₁ and y=m₂x+c₂ are distinct nonvertical parallel lines, then m₁=m₂ and c₁≠c₂.

Equal gradients alone do not prove the lines are distinct; identical equations describe the same line.

2. Perpendicular gradients are negative reciprocals

For nonvertical lines with finite nonzero gradients, perpendicularity is equivalent to m₁m₂=−1.

Thus a line of gradient 3 is perpendicular to a line of gradient −1/3.

3. Special case: horizontal and vertical

A horizontal line has gradient 0. A vertical line has undefined gradient. They are perpendicular, even though the product rule cannot be used.

This is why “negative reciprocal” should be treated as a rule for finite nonzero gradients, not a universal formula covering every line.

4. Worked parallel-line equation

Find the line through (2,5) parallel to 3x−y+7=0.

Rearrange the given line: y=3x+7, so the gradient is 3. The required line has y−5=3(x−2), hence y=3x−1.

5. Worked perpendicular-line equation

Find the line through (4,−2) perpendicular to y=2x+1.

The perpendicular gradient is −1/2. Therefore y+2=−(1/2)(x−4), giving y=−x/2.

6. Prove a right angle from coordinates

Let A(0,0), B(4,2), C(3,4). Gradient AB=2/4=1/2. Gradient BC=(4−2)/(3−4)=−2. Their product is −1, so AB is perpendicular to BC and ∠ABC=90°.

7. Prove parallelism from coordinates

For A(1,2), B(5,6), C(0,4), D(3,7), gradient AB=1 and gradient CD=1. Since the lines have the same gradient, they are parallel unless they lie on the same line.

Check a point if necessary to distinguish coincident from distinct lines.

8. General-form shortcut

For Ax+By+C=0 with B≠0, gradient is −A/B. Therefore 2x+5y−1=0 has gradient −2/5.

A perpendicular nonvertical line has gradient 5/2.

9. Common mistakes

• Using opposite sign but not reciprocal for perpendicular gradients.
• Applying m₁m₂=−1 to a vertical line.
• Assuming equal gradients always mean two distinct parallel lines.
• Reading gradient directly from general form without rearranging or using −A/B.

10. Practice

1. State the gradient of a line parallel to y=−4x+3.
2. State the gradient of a line perpendicular to y=(1/3)x−2.
3. Find the line through (1,5) parallel to 2x+y=7.
4. Find the line through (2,−1) perpendicular to y=−3x+4.
5. Explain why x=5 and y=2 are perpendicular.

11. Answers

1. −4.
2. −3.
3. 2x+y=7 has gradient −2. y−5=−2(x−1), so y=−2x+7.
4. Perpendicular gradient 1/3. y+1=(1/3)(x−2).
5. x=5 is vertical and y=2 is horizontal.

Continue with Straight Lines in Additional Mathematics, the Coordinate Geometry guide, or return to the Additional Mathematics Hub.