6.1PureCore

Midpoints and Perpendicular Bisectors

Before tackling circles we need two coordinate-geometry tools: the midpoint of a line segment, and the perpendicular bisector — the line that cuts a segment exactly in half at a right angle. Both lead directly to finding the centre of a circle.

25 min Video by Zeeshan Zamurred Circles

Edexcel AS Level Maths: 6.1 Midpoints and Perpendicular BisectorsWatch the full walkthrough before the notes below.

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What you'll be able to do

Find the midpoint of two points
Find the gradient of the perpendicular to a line
Find the equation of a perpendicular bisector
Apply these to circle problems

The midpoint

The midpoint of two points is simply the average of their coordinates.

M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Average the

x

-coordinates and the

y

-coordinates separately.

1Average

x

(2 + 8) /2 = 5

2Average

y

(3 + 7) /2 = 5

Answer

(5, 5)

Perpendicular gradients

Two perpendicular lines have gradients that multiply to $- 1$ . So the perpendicular gradient is the $negative reciprocal$ — flip the fraction and change the sign.

m_{1} \times m_{2} = - 1 \Rightarrow m_{2} = - \frac{1}{m _{1}}

Negative reciprocal: e.g. gradient

2 \to - \frac{1}{2}

The perpendicular bisector

The perpendicular bisector passes through the $midpoint$ of a segment and is $perpendicular$ to it. Find the midpoint, find the perpendicular gradient, then use $y - y_{1} = m (x - x_{1})$ .

1Midpoint:

(3, 4)

2Gradient of

A B

\frac{6 - 2}{5 - 1} = 1

, so perpendicular gradient

= - 1

3Equation:

y - 4 = - 1 (x - 3)

Answer

y = - x + 7

Tip — The centre of a circle lies on the perpendicular bisector of any chord — this is the key link to circles.

Formula recap

M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Midpoint.

m_{1} m_{2} = - 1

Perpendicular gradients.

y - y_{1} = m (x - x_{1})

Line through a point.

Common mistakes to avoid

Using the reciprocal but forgetting the negative sign.

The perpendicular gradient is the NEGATIVE reciprocal: 2 → −½.

Using one endpoint instead of the midpoint for the bisector.

The perpendicular bisector passes through the midpoint of the segment.

Key takeaways

Midpoint = average of the coordinates.
Perpendicular gradient = negative reciprocal (m₁m₂ = −1).
Perpendicular bisector goes through the midpoint, perpendicular to the segment.
A circle’s centre lies on the perpendicular bisector of any chord.

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