Bisector = midpoint + perpendicular: A perpendicular bisector of a segment cuts it exactly in half at a right angle.
So it does two things at once: it passes through the midpoint of the two endpoints, and it has the negative-reciprocal gradient of the segment.
Picture it: with A(1, 2) and B(5, 8), the perpendicular bisector passes through the midpoint M(3, 5) and crosses AB at a right angle — and every point on it is the same distance from A as from B.
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Every point on it is equidistant: A neat way to picture it: the perpendicular bisector of AB is the set of all points the same distance from A as from B.
That's why it appears in 'equally far from two places' problems.
The three-step recipe
- 1. Midpoint of the two endpoints — average the coordinates.
- 2. Gradient of the segment, then its negative reciprocal (flip and change the sign).
- 3. Put that gradient and the midpoint into y = mx + c, then solve for c.
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Three steps, in order: Midpoint → negative-reciprocal gradient → line through the midpoint.
One small rule per step.
IB-style question — perpendicular bisector of two points
Find the equation of the perpendicular bisector of A(1, 2) and B(5, 8).
Step by step
- Midpoint formula, then substitute A and B.
- Gradient formula, then substitute.
- Bisector gradient = negative reciprocal.
- Substitute the midpoint (3, 5) into y = mx + c.
Final answer
y = −⅔x + 7.
IB-style question — give the answer in general form
Find the perpendicular bisector of P(−2, 1) and Q(4, 5), giving your answer as ax + by + d = 0.
Step by step
- Midpoint formula, then substitute P and Q.
- Gradient formula, then negate-and-flip.
- Through M(1, 3): 3 = −3⁄2(1) + c ⇒ c = 9⁄2.
- Clear fractions and move to one side.
Final answer
3x + 2y − 9 = 0.
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Spot this shape — the 3-part bisector question: Two points, then 'the line perpendicular to the segment through its midpoint' — the perpendicular bisector, asked in three quick parts.
Same three moves every time:
(a) Midpoint
- average the two points
- → the point M
(b) Gradient
- gradient of the segment
- → negative reciprocal
(c) Equation
- that gradient through M
- → y = mx + c
Step through it with the buttons: plot A and B → gradient of AB → midpoint M → perpendicular direction → the bisector.
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Part (a) — find the midpoint M
Point A has coordinates (1, 2) and point B has coordinates (7, 6).
Point M is the midpoint of [AB].
Find the coordinates of M.
Step by step
- Midpoint formula — average each coordinate.
- Substitute A(1, 2) and B(7, 6).
Final answer
M = (4, 4).
Part (b) — find the gradient of L
Line L is perpendicular to [AB] and passes through M.
Find the gradient of L.
Step by step
- Gradient formula for the segment [AB].
- Substitute A(1, 2) and B(7, 6).
- Perpendicular ⇒ negative reciprocal (flip and change the sign).
Final answer
gradient of L = −3/2.
Part (c) — write down the equation of L
Hence, write down the equation of L.
Step by step
- Put the gradient −3/2 through the midpoint M(4, 4) into point–gradient form.
- Tidy to y = mx + c.
Final answer
y = −3/2 x + 10 (or leave it as y − 4 = −3/2(x − 4)).
See how every part is linked: (a) midpoint → (b) perpendicular gradient → (c) line through M. Part (c) just drops the gradient from (b) and the point from (a) into point–gradient form — so three marks come quickly if you go in order.
Two marks people throw away: 1. The bisector passes through the midpoint M — not through A or B.
2. Use the negative reciprocal (flip and change the sign); point–gradient form like y − 4 = −3/2(x − 4) is accepted, but not 'L = …'.
Same midpoint skill, a different finish: Coordinate-geometry questions often open with a midpoint, then ask for the equation of a line through two points (here, not perpendicular).
Two linked moves:
(a) Midpoint
- average the two points
- → the point M
(b) The line
- gradient between two points
- → y = mx + c through one of them
Part (a) — find the midpoint M
Points A(8, 0) and C(2, 6) are two corners of a shape.
M is the midpoint of [AC].
Find the coordinates of M.
Step by step
- Midpoint formula — average each coordinate.
- Substitute A(8, 0) and C(2, 6).
Final answer
M = (5, 3).
Part (b) — the line through O and M
Hence find the equation of the line through the origin O(0, 0) and M.
Step by step
- Gradient formula between the two points.
- Substitute O(0, 0) and M(5, 3).
- Through the origin the y-intercept is 0, so y = mx.
Final answer
y = (3/5)x.
See how the parts link: Part (a) hands you a point; part (b) uses that point with the origin to get a gradient, then the line.
A line through the origin always has the tidy form y = mx (c = 0).