Question

two tangents PQ and PR are drawn from an external point P to the circle with radius O prove that .PROQ is a cyclic quadrilateral

Answer 1

Since tangent at a point to a circle is perpendicular to the radius through the point.

OP ⊥ QP and OR ⊥ RP.

∠OQP = 90° and ∠ORP = 90°

∠OQP + ∠ORP =( 90° + 90° )= 180°

∠OQP + ∠ORP = 180°……………(1)

In quadrilateral OQPR

∠OQP + ∠QPR+ ∠QOR + ∠ORP = 360°

(∠QPR + ∠QOR)+ (∠OQP + ∠ORP) = 360°

(∠QPR + ∠QOR) + 180° = 360° [ From (1) ]

∠QPR + ∠QOR = 180° ................ (2)

From (1) and (2), we can say that the sum of a pair of opposite angles of a quadrilateral QORP is 180° .

Hence, quadrilateral QORP is cyclic quadrilateral