Q) 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 contacts at the centre.


prove that the angle between CBSE 10th Board important PYQ

Let’s draw a diagram with a circle of radius r and O as centre. Let the two tangents RP & RQ are drawn from point R on to this circle, The tangent RP touches the circle at point P and tangent RQ touches the circle at point Q.

We need to prove that, QRP  + POQ  = 180°

From  the diagram, we can see that RQOP forms a quadrilateral and we know that, the sum of all 4 angles in any quadrilateral is 3600.

\therefore QRP +  RPO + POQ + RQO  = 360° ……. (i)

Since we know that a tangent is always perpendicular to the radius, therefore

RPO = RQO = 90°

Substituting these values in equation (i), we get

QRP + 90° + POQ + 90°  = 360°

QRP  + POQ  = 180° …… Hence Proved!

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