Given: TS is a tangent to the circle with centre O at P, and OP is joined.
Required to prove: OP is perpendicular to TS which passes through the centre of the circle
Construction: Draw a line OR which intersect the circle at Q and meets the tangent TS at R.
Proof:
OP = OQ (radii of the same circle)
And OQ < OR
⇒ OP < OR
similarly, we can prove that OP is less than all lines which can be drawn from O to TS.
OP is the shortest
OP is perpendicular to TS
Therefore, the perpendicular through P will pass through the centre of the circle
– Hence proved.
