Q. 29

# Prove that, a tan

Answer :

Let us consider a circle with center O, OP be a radius and XY be a tangent at point P. To prove: OP XY

Proof: Let’s take a point Q on XY other than P.

Clearly, point Q lies outside the circle because if Q lies inside the circle then XY will be a secant (as it will intersect the circle at two points)

Point P is on circle and point Q is outside the circle

OP < OQ

This is true for all points on XY other than P

OP is the shortest distance between point P and line XY.

OP XY [Shortest side is perpendicular]

Hence, proved!

OR Let us consider a circle with center O. PQ and PR be two tangents from an external point P to the circle.

To prove: QPR + QOR = 180°

Proof: In quadrilateral PROQ, we have

QPR + PQO + QOR + ORP = 360°

Now, OQP = ORP = 90° [Tangent at a point on a circle is perpendicular to the radius through point of contact]

⇒∠QPR + 90° + QOR + 90° = 360°

⇒∠QPR + QOR = 180°

Hence, proved!

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