Locus: perpendicular bisector
Points equidistant from A and B lie on the perpendicular bisector of AB.
Key formulas
Geometry meaning
Set distance to A equal to distance to B and expand. The x² and y² terms cancel, leaving the equation of a line.
Example
A(0,0), B(4,0). Midpoint (2,0), line x=2.
Construction
In diagrams without coordinates, construct the midpoint and erect a perpendicular.
Worked example
Worked example
Problem. Find the locus of P(x, y) such that PA = PB for A(0,0) and B(4,0).
Solution. √(x²+y²) = √((x−4)²+y²). Square and simplify: 0 = −8x + 16 → x = 2. The locus is the line x = 2.
Circle link
Intersection of two perp bisectors of chords gives a circle centre.
Review quiz
Points equidistant from (1,1) and (5,1) lie on:
Practice questions
- A(1,1), B(5,1). Write the locus |PA|=|PB|.
- Why do squares cancel in the algebra?
Study notes
Is the midpoint on the locus?
Yes — it is equidistant from A and B.