RD Sharma Class 10 Solutions Chapter 10 Circles Ex 10.2 Q10

### Q10.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 points of contact at the centre.

In order 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 points of contact at the centre, we will use the following theorem:

Theorem: If a line is drawn parallel to a chord of a circle, it will bisect the angle subtended by the chord.

Proof:

Let ABC be a chord of a circle with centre O. Let D be the point on the circle that is external to the chord ABC. Let E be the point on the line parallel to the chord ABC that is closest to the point D.

Since E is the point on the line parallel to the chord ABC that is closest to the point D, it follows that E is the point on the line parallel to the chord ABC that is closest to the centre of the circle.

Since E is the point on the line parallel to the chord ABC that is closest to the centre of the circle, it follows that the angle EOD is equal to the angle ABC.

It also follows that the angle BOD is equal to the angle ACD.

Since the angles BOD and ACD are supplementary, it follows 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 points of contact at the centre.