Bisection of Chords (Cambridge International AS Mathematics)

Bisection of chords

How can I find the equation of a perpendicular bisector?

• The perpendicular bisector of a line segment:

  • is perpendicular to the line segment
  • goes through the midpoint of the line segment

• The midpoint and gradient of the line segment between points (x₁, y₁) and (x₂, y₂) are given by the formulae

• The gradient of the perpendicular bisector is therefore

• The equation of the perpendicular bisector is the equation of the line with that gradient through the line segment's midpoint (see Equation of a Straight Line)

How can I use perpendicular bisectors to find the equation of a circle?

• A chord of a circle is a straight line segment between any two points on the circle

• The perpendicular bisector of a chord always goes through the centre of the circle

• If you know three points on a circle, draw any two chords between them – the perpendicular bisectors of the chords will meet at the centre of the circle

• Now that you know the centre of the circle and a point on the circle you can write the equation of the circle

Exam Hint

• To find the point of intersection of two straight lines, set the equations of the lines equal to each other and solve.

Worked Example