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, \(\angle OBA=90{^\circ}\)
\(\begin{aligned}\\ \angle AOB + 138{^\circ}&= 180 {^\circ} \\\\\angle AOB&= 180 {^\circ} - 138{^\circ} \\\\&=42{^\circ}.\\\\ \end{aligned}\)
\(\begin{aligned} x + \angle AOB&= 90 {^\circ} \\\\x&= 90{^\circ} - \angle AOB \\\\x&=90{^\circ} - 42{^\circ} \\\\x&=48.{^\circ} \end{aligned}\)
If two tangents to a circle with centre \(O\) and points of tangency \(B\) and \(C\) meet at point \(A\), then
\(\angle x=\angle y\) 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.
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