To find the equation of a perpendicular bisector, use the midpoint of the segment and a gradient perpendicular to the segment. For a sloping segment with gradient m, the bisector has gradient −1/m. Horizontal and vertical segments use a constant-coordinate equation instead.
The 2027 SEC G3 Mathematics K310 syllabus covers perpendicular bisectors and coordinate geometry. This guide combines those ideas and shows two ways to check the equation.
- Find the midpoint by averaging each coordinate separately.
- Calculate the gradient of the original segment.
- Use the perpendicular gradient and the midpoint to write the equation.
- Check the midpoint and the right-angle condition separately.
CHAPTER 1 OF 4 · Back to contents
Find the point and direction separately
The bisector crosses the segment at its midpoint. Average the two x-coordinates to find its x-coordinate, then average the two y-coordinates to find its y-coordinate.
The direction comes from the perpendicular condition. For a segment whose gradient is finite and non-zero, multiply the two gradients to check that their product is −1.
A line through the midpoint can still have the wrong slope. A line with the right slope can still miss the midpoint. Keep both checks in your working.
CHAPTER 2 OF 4 · Back to contents
Worked example | Bisect the segment joining (2, 1) and (8, 5)
The midpoint is ((2 + 8)/2, (1 + 5)/2) = (5, 3). From the first point to the second, y rises by 4 and x increases by 6, so the gradient is 4/6 = 2/3.
| Item | Result | How it is used |
|---|---|---|
| Midpoint | (5, 3) | A point on the bisector |
| Segment gradient | 2/3 | Find the perpendicular gradient |
| Bisector gradient | −3/2 | Use with the midpoint |
Using the midpoint gives y − 3 = −(3/2)(x − 5). This simplifies to y = −(3/2)x + 21/2. Substituting x = 5 returns y = 3, and (2/3)(−3/2) = −1.
Another check is to use the y-intercept (0, 21/2). The squared distances from it to both endpoints are 377/4. This agrees with the property that every point on the perpendicular bisector is equally far from both endpoints.
Contents · Previous chapter · Next chapter
CHAPTER 3 OF 4 · Back to contents
Practise with a horizontal or vertical segment
The points (−2, 4) and (6, 4) make a horizontal segment. Its midpoint is (2, 4). The perpendicular bisector is vertical, so its equation is x = 2.
The points (3, −1) and (3, 7) make a vertical segment. Its midpoint is (3, 3). The perpendicular bisector is horizontal, so its equation is y = 3.
Sketch each segment and ask which coordinate stays fixed along its bisector. This avoids trying to divide by zero in the horizontal case or assigning a finite gradient to a vertical line.
Contents · Previous chapter · Next chapter
CHAPTER 4 OF 4 · Back to contents
Frequently asked questions
Can I leave the answer in point-gradient form?
Yes if the question allows any equivalent equation. If a form is requested, rearrange to that form.
What if both points have the same coordinates?
They do not determine a segment with a direction. A unique perpendicular bisector requires two distinct endpoints.
What is a quick sign check?
A rising segment has a falling perpendicular line, and a falling segment has a rising perpendicular line.
Contents · Previous chapter · Continue with the related guide