## Tangents to Circles

6.3  Tangents to Circles

 Definition Tangent to a circle is a straight line that touches the circle at only one point. The point of contact between the tangent and the circle is the point of tangency.

The value of the angle between tangent and radius at the point of tangency:

• The radius of a circle that intersects with the tangent to the circle at the point of tangency will form a $$90{^\circ}$$ angle with the tangent.

 Example The following diagram shows a circle with centre $$O$$ which meets the straight line $$ABC$$ at point $$B$$ only. Calculate the value of $$x$$. Line $$ABC$$ is a tangent to the circle and it touches the circle at point $$B$$. So, $$\angle OBA=90{^\circ}$$  \begin{aligned}\\ \angle AOB + 138{^\circ}&= 180 {^\circ} \\\\\angle AOB&= 180 {^\circ} - 138{^\circ} \\\\&=42{^\circ}.\\\\ \end{aligned}        \begin{aligned} x + \angle AOB&= 90 {^\circ} \\\\x&= 90{^\circ} - \angle AOB \\\\x&=90{^\circ} - 42{^\circ} \\\\x&=48.{^\circ} \end{aligned}

Properties of two tangents to a circle:

If two tangents to a circle with centre $$O$$ and points of tangency $$B$$ and $$C$$ meet at point $$A$$, then

• $$\angle BA=\angle CA$$
• $$\angle BOA=\angle COA$$
• $$\angle OAB=\angle OAC$$

The relationship of the angle between tangent and chord with the angle in the alternate segment which is subtended by the chord:

• $$\angle x=\angle y$$ and $$\angle \theta=\angle \beta$$ because the angles between the chords and the tangents are equal to the angles at the alternate segments subtended by the chords.

Common tangents:

• A common tangent to two circles is a straight line that is a tangent to both circles.