AQA A Level Maths: Pure复习笔记3.2.3 Bisection of Chords

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

 

3.2.3-Perpendicular-Bisector

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

3.2.3-Midpoint-Gradient-Forms

  •  The gradient of the perpendicular bisector is therefore

3.2.3-Perp-Bisector-Gradient

  • 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

3.2.3-Chord

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

3.2.3-Perp-Bisect-Chord

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

3.2.3-Perp-Bisect-Centre

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

Exam Tip

  • To find the point of intersection of two straight lines, set the equations of the lines equal to each other and solve. 3.2.3-Find-lines-intersect

Worked Example

3.2.3-Bisect-Chords-Example

转载自savemyexams

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