Q. 84.0( 2 Votes )

# Rumela drew a circle with centre with centre O of which QR is a chord. Two tangents drawn at the points Q and R intersect each other at the point P. If QM is a diameter, let us prove that ∠QPR = 2 ∠RQM.

Formula used.

Isosceles triangle property

If 2 angles of triangle are equal then their corresponding sides are also equal

Perpendicular drawn through tangent pass through centre

Solution

Join OR

As QP is tangent at point Q and RP tangent to point R

Hence;

OQP = ORP = 90°

In Δ OQR

As OQ = OR

By isosceles triangle property

OQR = ORQ

In Δ PQR

By angle sum property

P + PQR + PRQ = 180°

P + [90° - OQR] + [90° - ORQ] = 180°

P + OQR + ORQ = 180° - 180°

P = OQR + ORQ

P = 2OQR

P = 2MQR

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