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.

Ans:

Step 1: 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°

Step 2: From  the diagram, we can see that RQOP forms a quadrilateral

∵  Sum of all 4 angles in any quadrilateral is 360⁰.

∴ ∠ QRP + ∠ RPO + ∠ POQ + ∠ RQO  = 360° ……. (i)

Step 3: Since we know that a tangent is always perpendicular to the radius

∴ ∠ RPO = ∠ RQO = 90°

Step 4: Substituting these values in equation (i), we get

∴ ∠ QRP + 90° + ∠ POQ + 90°  = 360°

∴ ∠ QRP  + ∠ POQ  + 180° = 360° 

∴ ∠ QRP  + ∠ POQ  = 180°   …… Hence Proved!

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