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Equation of a Perpendicular Bisector

Maths: Edexcel A Level Pure Maths, topic 3.2.7

Perpendicular Bisector of a Chord

We can find the equation of the perpendicular bisector of a chord using the properties of straight lines graphs.

What is the perpendicular bisector of a chord?

What is the perpendicular bisector of a chord?

  • The perpendicular bisector of a chord is the perpendicular line that passes through the midpoint of the chord.
  • The perpendicular bisector of a chord always passes through the centre of the circle.
What is the gradient of the perpendicular bisector?

What is the gradient of the perpendicular bisector?

  • We first need to find the gradient of the chord using the equation:
  • mc=y2−y1x2−x1m_c = \frac{y_2-y_1}{x_2-x_1}
  • Where (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2) are the points where the chord meets the circumference.
  • The bisector is perpendicular to the chord, so its gradient mbm_b satisfies:
  • mb×mc=−1m_b\times m_c=-1
What is the gradient of the perpendicular bisector?

What is the gradient of the perpendicular bisector?

What is the equation of the perpendicular bisector?

What is the equation of the perpendicular bisector?

  • We can then find the equation of the bisector by working out the midpoint of the chord:
  • Midpoint = (x1+x22,y1+y22)(\frac{\small x_1+x_2}{2},\frac{\small y_1+y_2}{2})
  • Finally, we substitute this into the general equation of a straight line with gradient mbm_b to find the yy-intercept cc:
  • y=mbx+cy = m_bx + c
Example

Example

  • The points A(3,7)A(3,7) and B(6,4)B(6,4) lie on the circumference of a circle.
  • What is the equation of the perpendicular bisector of the chord AB?
Find the gradient of chord

Find the gradient of chord

  • The gradient of the chord is given by:
  • mc=y2−y1x2−x1=4−76−3=−33=−1m_c = \frac{y_2-y_1}{x_2-x_1}=\frac{4-7}{6-3}=-\frac{3}{3}=-1
Find the gradient of bisector

Find the gradient of bisector

  • The gradient of the perpendicular bisector mbm_b satisfies the equation:
  • mb×mc=−1m_b\times m_c = -1
  • mb=−1mc=1m_b = -\frac{1}{m_c} = 1
Find the midpoint of chord

Find the midpoint of chord

  • The midpoint of the chord AB is equal to:
  • midpoint = (x1+x22,y1+y22)=(3+62,7+42)=(4.5,5.5)(\frac{\small x_1+x_2}{2},\frac{\small y_1+y_2}{2}) = (\frac{3+6}{2},\frac{7+4}{2}) = (4.5, 5.5)
Find the y-intercept of the perpendicular bisector

Find the y-intercept of the perpendicular bisector

  • Substitute mbm_b and the midpoint into the general equation of a straight line:
  • y=mbx+cy = m_bx+c
  • 5.5=1×4.5+c5.5 = 1\times 4.5 + c
  • Rearranging for cc, we get:
  • c=5.5−4.5=1c = 5.5 - 4.5 =1
Final answer

Final answer

  • So the equation of the perpendicular bisector of the chord AB is:
  • y=x+1y =x + 1

Test yourself on Equation of a Perpendicular Bisector

  • Where does the perpendicular bisector of a chord intersect the chord?

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