Two properties in one line: The perpendicular bisector of segment AB: (1) passes through the midpoint of AB, and (2) is perpendicular to AB.
Every point on the bisector is equidistant from A and B.
[Diagram: math-perpendicular-bisector] - Available in full study mode
Connection to Voronoi: In a Voronoi diagram, the boundary between two cells is the perpendicular bisector of the two sites.
This is why perpendicular bisectors are foundational to 3.6.
Three-step method: Step 1: Find the midpoint M.
Step 2: Find the gradient of AB, then take the negative reciprocal for the perpendicular gradient.
Step 3: Use y − y₁ = m(x − x₁) with M as the point.
Worked example
Find the equation of the perpendicular bisector of AB where A = (1, 3) and B = (5, 7).
Step by step
- Step 1: Midpoint M.
- Step 2: Gradient of AB.
- Perpendicular gradient (negative reciprocal).
- Step 3: Equation through M(3, 5) with m = −1.
Final answer
Perpendicular bisector: y = −x + 8.
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Perpendicular gradients multiply to −1: If two lines are perpendicular, their gradients m₁ and m₂ satisfy: m₁ × m₂ = −1.
So if AB has gradient 3, the perpendicular has gradient −1/3.
| Gradient of AB | Perpendicular gradient |
|---|---|
| 2 | −1/2 |
| −3 | 1/3 |
| 1/4 | −4 |
| 0 (horizontal) | Undefined (vertical line) |
Zero gradient special case: If AB is horizontal (gradient 0), the perpendicular bisector is a vertical line x = midpoint x-value.
Worked example — verify a point lies on the bisector
The perpendicular bisector of PQ, where P = (2, 4) and Q = (6, 2), passes through the point (4, 6).
Verify this.
Step by step
- Find bisector equation. Midpoint: M = (4, 3). Gradient PQ = (2−4)/(6−2) = −1/2. Perpendicular gradient = 2.
- Equation: y − 3 = 2(x − 4) → y = 2x − 5.
- Check (4, 6): y = 2(4) − 5 = 3 ≠ 6.
Final answer
The point (4, 6) does NOT lie on the bisector (y = 2x − 5 gives y = 3 at x = 4, not 6).
IB-style question — equidistant point on a road
Two shops are at A(2, 1) and B(8, 5). A new café must be the same distance from both shops and must lie on the road y = x − 1.
Find the coordinates of the café.
Step by step
- 'Same distance from A and B' means the café is on the perpendicular bisector of [AB]. Midpoint and gradient first.
- Equation of the perpendicular bisector through M.
- It must also be on the road, so set the two lines equal and solve.
Final answer
The café is at (4.6, 3.6).
[Diagram: math-perpendicular-bisector] - Available in full study mode