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.

Answer :

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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