PA and PB are the tangents to a circle which circumscribes an equilateral triangle Triangle ABQ If angle PAB - Maths - Circles

Question

PA and PB are the tangents to a circle which circumscribes an
equilateral triangle Triangle ABQ If angle PAB = 60 degree , prove
that QP bisects AB at right angles

Somnath maths · Accepted Answer

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Given,ΔAQB is equilateral
 PA and PB are tangents drawn from P
To prove: PQ perpendicular bisector of AB
Proof:We have in ΔPAQ and ΔPBQAQ = QB             [∵sides of equilateral triangle are equal]PQ = PQ              [∵ common side]PA = PB              [∵ Tangents drawn from a point to a circle are equal]ΔPAQ ≅ ΔPBQ [∵SSS SSS congruence condition]∠PQA=∠PQB    [∵CPCT]In ΔACQ and ΔBCQAQ = QB [∵sides of equilateral triangle are equal]∠PQA=∠PQB CQ = CQ [∵common side]ΔACQ ≅ ΔBCQ AC = CB [∵CPCT]Thus, QC median of equilateral ΔAQB
QC⊥AB [∵Median of an equilateral triangle is its perpendicular bisector ]So QP is perpendicular bisector of ABProved

PA and PB are the tangents to a circle which circumscribes an equilateral triangle Triangle ABQ. If angle PAB = 60 degree , prove that QP bisects AB at right angles