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Two tangent TP and TQ are drawn to a circle with centre O from an external point T. Prove that <PTQ = 2 angle OPQ

Posted by Manisha Choudhary 2 years ago

CBSE > Class 10 > Mathematics

1 answers

Preeti Dabral 2 years ago

Given A circle with centre O and an external point T and two tangents TP and TQ to the circle, where P, Q are the points of contact.
To Prove: $\angle$ PTQ = 2 $\angle$ OPQ
Proof: Let $\angle$ PTQ = $\theta$
Since TP, TQ are tangents drawn from point T to the circle.
TP = TQ
$\therefore$ TPQ is an isoscles triangle
$\therefore$ $\angle$ TPQ = $\angle$ TQP = $\frac12$ (180^o - $\theta$ ) = 90^o - $\fracθ2$
Since, TP is a tangent to the circle at point of contact P
$\therefore$ $\angle$ OPT = 90^o
$\therefore$ $\angle$ OPQ = $\angle$ OPT - $\angle$ TPQ = 90^o - (90^o - $\frac12$ $\theta$ ) = $\fracθ2$ = $\frac12$ $\angle$ PTQ
Thus, $\angle$ PTQ = 2 $\angle$ OPQ

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CBSE > Class 10 > Mathematics