Question

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the point of contact at the center.

Answer 1

Hint: In this given question, we must, first of all, construct a suitable diagram according to the information given in the question. Then we may prove the given statement by using the fact that the angles formed by radii and tangents are always a right angle and the sum of all the angles of a quadrilateral is ${{360}^{\circ }}$.

Complete step-by-step solution -
In this given question, we are asked to prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the point of contact at the center.
Let us first construct a suitable diagram according to the given information.

Here, AB and CB are two tangents from point B to, a circle with center O.
Now, as we know that the angles formed by radii and tangents are always a right angle, we can conclude that $\angle OAB=\angle OCB={{90}^{\circ }}$ from this fact.
$\angle OAB=\angle OCB={{90}^{\circ }}..............(1.1)$
As per the angle sum property, the sum of all the angles of a quadrilateral is ${{360}^{\circ }}$.
So, from the above statement, we get,
$\angle OAB+\angle ABC+\angle OCB+\angle AOC={{360}^{\circ }}...............(1.2)$
By putting values of angles obtained from 1.1 in 1.2, we get,
\[\begin{align}
  & \angle OAB+\angle ABC+\angle OCB+\angle AOC={{360}^{\circ }} \\
 & \Rightarrow {{90}^{\circ }}+\angle ABC+{{90}^{\circ }}+\angle AOC={{360}^{\circ }} \\
 & \Rightarrow \angle ABC+\angle AOC={{360}^{\circ }}-{{180}^{\circ }}={{180}^{\circ }}...........(1.3) \\
\end{align}\]
Now, we know that two angles are supplementary if their sum is ${{180}^{\circ }}$.
Hence, from equation 1.3, we get \[\angle ABC\] and \[\angle AOC\] as supplementary angles.
Therefore, the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the point of contact at the center.

Note: In this question, we should be careful to use the fact that the angle between a tangent and the line joining the point of intersection to the center of the circle should be a right angle because only then we can use the property that the sum of angles of a quadrilateral is ${{360}^{\circ }}$ and prove that the angles specified in the question are supplementary.