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 Circles

Edexcel AS Level Maths: 6.1 Midpoints and Perpendicular BisectorsCurated by Kepler, not produced by us. Watch it 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 = \left( \dfrac{x_1 + x_2}{2}, \; \dfrac{y_1 + y_2}{2} \right)

Average the

x

-coordinates and the

y

-coordinates separately.

Example 1Find the midpoint of

A(2, 3)

and

B(8, 7)

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 $\textbf{negative reciprocal}$ — flip the fraction and change the sign.

m_1 \times m_2 = -1 \;\Rightarrow\; m_2 = -\dfrac{1}{m_1}

Negative reciprocal: e.g. gradient

2 \to -\tfrac{1}{2}

The perpendicular bisector

The perpendicular bisector passes through the $\textbf{midpoint}$ of a segment and is $\textbf{perpendicular}$ to it. Find the midpoint, find the perpendicular gradient, then use $y - y_1 = m(x - x_1)$ .

Example 1Find the perpendicular bisector of

A(1, 2)

and

B(5, 6)

1Midpoint:

(3, 4)

2Gradient of

AB

\tfrac{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 = \left( \tfrac{x_1+x_2}{2}, \tfrac{y_1+y_2}{2} \right)

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.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.