Circles Tangents Problems

Problem 1

Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ

Interpretation of the problem

1. O is the centre of the circle and tangents AP and AQ are drawn from an external point A. #OP and OQ are the radii. #The students have to prove thne angle PAQ=twise the angle OPQ.

Concepts used

1. The radii of a circle are equal.
2. In any circle the radius drawn at the point of contact is perpendicular to the tangent.
3. The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.
4. Properties of isoscles triangle.
5. Properties of quadrillateral ( sum of all angles) is 360 degrees
6. Sum of three angles of triangle is 180 degrees.

Algorithm

OP=OQ ---- radii of the same circle OA is joined.
In quadrillateral APOQ ,
∠APO=∠AQO=$90^{0}$ [radius drawn at the point of contact is perpendicular to the tangent]
∠PAQ+∠POQ=$180^{0}$
Or, ∠PAQ+∠POQ=$180^{0}$
∠PAQ = $180^{0}$-∠POQ ----------1
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP
∠POQ+∠OPQ+∠OQP=$180^{0}$
Or ∠POQ+2∠OPQ=$180^{0}$
2∠OPQ=$180^{0}$- ∠POQ ------2
From 1 and 2
∠PAQ=2∠OPQ