6.4PureStretch

Use Tangent and Chord Properties

Circles come with elegant geometric facts. The two that matter most here: a tangent is perpendicular to the radius at the point of contact, and the perpendicular from the centre to a chord bisects it. Together they solve a huge range of problems.

30 min Video by Zeeshan Zamurred Circles

Edexcel AS Level Maths: 6.4 Use Tangent and Chord PropertiesWatch the full walkthrough before the notes below.

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What you'll be able to do

Use the tangent–radius perpendicularity property
Find the equation of a tangent to a circle
Use the fact that the centre lies on a chord’s perpendicular bisector
Solve circle geometry problems combining these facts

Tangent ⟂ radius

At the point where a tangent touches a circle, the tangent is $perpendicular$ to the radius drawn to that point. This lets you find the gradient of a tangent from the radius gradient.

m_{tangent} = - \frac{1}{m _{radius}}

Negative reciprocal of the radius gradient.

1Radius gradient

O P = \frac{4}{3}

2Tangent gradient is the negative reciprocal:

- \frac{3}{4}

Answer

- \frac{3}{4}

Tip — To get a tangent’s equation: find the radius gradient, take its negative reciprocal, then use y − y₁ = m(x − x₁) at the contact point.

The perpendicular from the centre bisects a chord

A line from the centre that is perpendicular to a chord cuts the chord exactly in half. Equivalently, the perpendicular bisector of any chord passes through the centre — a powerful way to find the centre.

Combining the properties

Many problems give two points on a circle (a chord) and ask for the centre or radius. Find the perpendicular bisector of each chord; the centre is where they meet. The radius is then the distance from the centre to any point on the circle.

Formula recap

m_{tangent} = - \frac{1}{m _{radius}}

Tangent ⟂ radius.

centre \in perp. bisector of any chord

Finding the centre.

r = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Radius via distance.

Common mistakes to avoid

Using the radius gradient itself as the tangent gradient.

The tangent is perpendicular: use the negative reciprocal.

Assuming the perpendicular from the centre meets the chord off-centre.

It bisects the chord — it meets it exactly at the midpoint.

Key takeaways

A tangent is perpendicular to the radius at the contact point.
Tangent gradient = negative reciprocal of the radius gradient.
The perpendicular from the centre bisects a chord; the centre lies on every chord’s perpendicular bisector.

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