Tangents to Circles

6.3  Tangents to Circles
  • 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.


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.