Question

Two circles touch externally at a point P. From a point T on the tangent at P, tangents TQ and TR are drawn to the circles with points of contact Q and R respectively. Prove that TQ = TR

Answer 1

Explanation:

Given that two circles touch externally at a point P.

Also, given that from the point T on the tangent at P, tangents TQ and TR are drawn to the circles with points of contact Q and R respectively.

We need to prove that TQ = TR

The image of the two circles and the tangent is attached below:

From the figure, we can see that the tangents TR and TP are tangents to one of the circles.

Then, by two tangent theorem, which states that "if two lines were drawn from the same point which lies outside a circle, such that both lines are tangent to the circle, then their lengths are the same".

Hence, we have,

$T R=T P$  -----------(1)

Similarly, the two tangents TP, TQ are tangents to another circle.

Then by two tangent theorem, we have,

$T P=T Q$  -------------(2)

Equating (1) and (2), we have,

$T Q=T R$

Hence proved.

Learn more:

(1) Two circles touch external point P. from a point T on the tangent at P, tangents TQ andTR

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(2) Two tangent TP and TQ are drawn to a circle with Centre O from an external point T. prove that angle PTQ =2angle OPQ.

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