Two lines with gradients m₁ and m₂ are perpendicular when m₁ × m₂ = −1. In words: the perpendicular gradient is the negative reciprocal. You flip the fraction and change the sign.
This skill belongs to straight lines and linearisation and returns in using a tangent perpendicular to the radius in circle work.
How do you use the rule?
There are two jobs. To test perpendicularity, find both gradients and check their product is −1. To build a perpendicular line, find the original gradient, take its negative reciprocal, and then use a point on the new line.
Examples: 2/3 becomes −3/2, 4 (which is 4/1) becomes −1/4, and −5 becomes 1/5.
A horizontal line (gradient 0) is perpendicular to a vertical line, which has no gradient, and that case is the only exception to the rule.
A perpendicular bisector of a segment is the line that is perpendicular to it and passes through its midpoint. So you need the midpoint, the gradient of the segment, and the negative reciprocal.
Worked example
Find the equation of the perpendicular bisector of A(−2, 3) and B(4, 7).
Step 1, midpoint: ((−2 + 4) ÷ 2, (3 + 7) ÷ 2) = (1, 5).
Step 2, gradient of AB: (7 − 3) ÷ (4 − (−2)) = 4 ÷ 6 = 2/3.
Step 3, perpendicular gradient: flip and change the sign, so −3/2.
Step 4, equation through (1, 5): y − 5 = −(3/2)(x − 1). Multiply by 2: 2y − 10 = −3x + 3, so 3x + 2y = 13.
Check 1: the midpoint (1, 5) gives 3 + 10 = 13, so it lies on the line.
Check 2: take the point (3, 2) on the line, since 9 + 4 = 13. Then distance² to A is 5² + (−1)² = 26, and to B is (−1)² + (−5)² = 26. Equal, as a perpendicular bisector requires.
The mistake to watch for
A common slip is to do only half of the negative reciprocal.
Mistaken working: gradient of AB is 2/3, so the perpendicular gradient is 3/2 (flipped only) or −2/3 (negated only)
Test: (2/3) × (3/2) = 1, and (2/3) × (−2/3) = −4/9. Neither equals −1.
The correction is to always multiply the two gradients as a check. With −3/2: (2/3) × (−3/2) = −1, which is what perpendicular means.
Check yourself
1. The line 2x + 3y = 6 is given. Find the equation of the line through (4, 1) that is perpendicular to it.
Show answer
Rearrange: 3y = −2x + 6, so y = −(2/3)x + 2 and the gradient is −2/3. The perpendicular gradient is 3/2.
y − 1 = (3/2)(x − 4), so y = 1.5x − 6 + 1.
y = 1.5x − 5. Check: at x = 4, y = 6 − 5 = 1.
2. Are y = 4x − 1 and x + 4y = 8 perpendicular?
Show answer
The first has gradient 4. For the second, 4y = −x + 8, so y = −(1/4)x + 2 and the gradient is −1/4.
Product = 4 × (−1/4) = −1. Yes, they are perpendicular.
3. The line kx + 2y = 7 is perpendicular to y = 3x + 1. Find k.
Show answer
For kx + 2y = 7, 2y = −kx + 7, so the gradient is −k/2. The other line has gradient 3.
(−k/2)(3) = −1, so −3k = −2 and k = 2/3. Check: gradient = −1/3, and (−1/3) × 3 = −1.
Where this leads next
If you need to sharpen how gradients are found first, revisit finding a line from intercept information. Then try the mixed practice set, which includes a perpendicular bisector and a right-angle check on a triangle.
Some students can apply the rule only when the equation is already in y = mx + c form. A teacher can practise other forms with you in online one-to-one Additional Mathematics tuition.