Circle Theorems: Tangents & Chords

Chords: What Are They?

A chord is just a straight line that joins two points on the edge of a circle.

All diameters are chords, but not all chords are diameters.

If two chords are the same length, they sit the same distance from the centre.

Circle Theorem:

A radius that bisects a chord cuts it at right angles

In other words:

• If a radius goes through the midpoint of a chord, it must be perpendicular to it
• And it works the other way too

Example: Using the Chord Theorem

A circle has centre $O$, radius 6 cm. Points $P$ and $Q$ lie on the circle, and angle $OPQ = 40\degree$. Find the length of chord $PQ$.

Step-by-step:

Mark in the radius: $OP = 6 cm$

Drop a perpendicular from $O$ to chord $PQ$, splitting it into two equal parts

Use trigonometry in triangle $OPQ$:

Let half the chord be $x$:

$$\cos(40\degree) = \frac{x}{6}$$

$$x = 6 \cos(40\degree) \approx 4.60 \text{ cm}$$

Double it:

$$PQ = 2x = 9.19 \text{ cm (3 s.f.)}$$

Worked Example

In a circle with centre $O$, a chord $AB$ is bisected by a radius at $90\degree$. If $OA = 5 \text{ cm}$ and angle $OAB = 60\degree$, find the full length of chord $AB$.

Solution:

Use trigonometry:

$$\cos(60\degree) = \frac{\text{half of } AB}{5} \Rightarrow \frac{x}{5} = 0.5 \Rightarrow x = 2.5m$$

So the full length of $AB = 2x = 5 \text{ cm}$

Tangents: What Are They?

A tangent is a straight line that just touches the circle at one point. It never enters the circle.

Circle Theorem:

A radius and a tangent meet at right angles

This means:

If you draw a radius to the point of contact with the tangent, the angle is always $90\degree$

Another Theorem:

Two tangents from a point outside the circle are equal in length

This creates a special shape – a kite:

The tangents are the same length

The radii are equal

Right angles form at the points where tangents meet the circle

Example: Tangents and Angles

A circle has centre $O$. Tangents from a point $T$ outside the circle meet the circle at points $R$ and $S$.

You are told:

$$\angle RTS = 25\degree$$

Find $\angle ROS$.

Step-by-step:

Add radii: $OR$ and $OS$

Mark the right angles where each radius meets the tangent

Use the quadrilateral $ORTS$

Angles in a quadrilateral add to $360\degree$:

$$\theta + 90 + 90 + 25 = 360 \Rightarrow \theta = 155\degree$$

So,

$$\boxed{\angle ROS = 155\degree}$$

Worked Example

Tangents from a point $P$ meet a circle at points $A$ and $B$. The radius of the circle is 10 cm, and the distance from $P$ to the centre is 26 cm. Find the length of one tangent.

Solution:

Draw triangle $OPA$

Use Pythagoras:

$$PA^2 = OP^2 - OA^2 = 26^2 - 10^2 = 676 - 100 = 576 \\  \Rightarrow PA = \sqrt{576} = 24 \text{ cm}$$

So the length of each tangent is:

$$\boxed{24 \text{ cm}}$$