IF FROM ANY POINT ON THE COMMON CHORD OF TWO INTERSECTING CIRCLES,TANGENTS BE DRAWN TO THE CIRCLES PROVE THAT THEY ARE CONGRUENT.....

Ref.10th R.D. Sharma cbse mathematics book, lesson 11(circles), exercise 11.2 question no.3

plz ans..

gopal.mohanty... , Meritnation Expert added an answer, on 11/2/11

Hi Cheral!

In order to prove your question we will use one property. It can be stated as **Let PT be a tangent to the circle from an exterior point P and a secant to the circle through P intersects the circle at points A and B where T is a point on the circle, then PT**^{2} = PA.PB. First of all I will prove this and use it to prove your question.

Let PT be a tangent to the circle from an exterior point P and a secant to the circle through P intersects the circle at points A and B where T is a point on the circle

Using Pythagoras theorem for ∆OPT

OT^{2} + PT^{2} = OP^{2}

⇒*r* ^{2} + PT^{2} = *r* ^{2} + PA.PB [using (2)]

⇒PT^{2} = PA.PB … (3)

Now, I will use this result to prove your question.

The information provided by you is represented diagrammatically as

Here, the circles intersect at point X and Y. A is a point on the line joining the points X and Y. AM and AN are the tangents drawn to the circles

You need to prove AM = AN

Using (3), it can be said that

AM^{2} = AX.AY and AN^{2} = AX.AY

Thus, AM^{2} = AN^{2}

⇒AM = AN

Hence, proved

Hope! This will help you.

Cheers!

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