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: ] - 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.
- Step 3: Equation through M(3,5) with m = −1.
Final answer
Perpendicular bisector: y = −x + 8.
Memorize terms 3x faster
Smart flashcards show you cards right before you forget them. Perfect for definitions and key concepts.
Try Flashcards Free7-day free trial • No card required
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).