Question

two tangents PT and PQ are drawn to a circle with centre O from an external point T. Prove that anglePTQ=2angleOPQ​​

Answer 1

Answer:

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Answer 2

PROVED

Step-by-step explanation:

given : two tangents PT and PQ are drawn to a circle with centre O from an external point T

To prove: $\angle PTQ = 2\angle OPQ$

proof :

we know that the length of the tangent drawn from an external point to a circle are equal

so , TP = TQ

$\angle TPQ = \angle TQP$ (angle opp to equal sides )

PT is the tangent and OP is the radius

so,

$\angle OPT = 90^\circ \\\\\angle OPQ + \angle TPQ = 90^\circ \\\\\angle TPQ = 90 - \angle OPQ ..(2)$

In triangle PTQ

$\angle TPQ + \angle TQP + \angle PTQ = 180^\circ \\\\\angle TPQ + \angle TPQ + \angle PTQ = 180 \\\\2\angle TPQ + \angle PTQ = 180\\\\2(90-\angle OPQ)+ \angle PTQ = 180\\\\180-2\angle OPQ +\angle PTQ = 180 \\\\\angle PTQ = 180 - 180 + 2\angle OPQ \\\\\angle PTQ = 2 \angle OPQ$

HENCE PROVED

#Learn more :

Two circles  intersect at two pts. A & B. XY is a tangent at pt. "P  prove that CD is parallel to the tangent XY.​

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