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Bisection of Chords (CIE A Level Maths: Pure 1)
Revision Note
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
Examiner Tip
- To find the point of intersection of two straight lines, set the equations of the lines equal to each other and solve.
Worked example
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