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## #1 2013-10-25 02:37:07

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

### tangents to a circle

pls. help me out ...

Q.A circle is touching the side BC of a ΔABC at P and touching AB and AC produced at Q and R respt. Prove that AQ=1/2 (perimeter of ΔABC).

friendship is tan 90°.

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## #2 2013-10-25 03:15:56

Nehushtan
Member
Registered: 2013-03-09
Posts: 915
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### Re: tangents to a circle

Let the centre of the circle be O.

Then we have OP = OQ (theyre radii of the circle) and ∠OPB =∠OQB = 90°. Hence BQ = BP.

Similarly CR = CP.

Therefore AB + BP = AQ = AR = AC + CP.

∴ AQ = AC + CP = ½(AC + CP + AC +CP) = ½(AB + BP + CP + AC) = ½ × perimeter.

Last edited by Nehushtan (2013-10-25 07:30:31)

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## #3 2013-10-25 03:57:13

bob bundy
Registered: 2010-06-20
Posts: 8,354

### Re: tangents to a circle

hi niharika_kumar

The distances from a point outside a circle to the points where the tangent touches the circle are equal.

So for example AQ = AR and BQ = BP

I have called some distances w, x, y and z.  Look at my diagram below for which.

By using the tangent rule above you will be able to do this quickly.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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## #4 2013-10-25 16:46:58

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

### Re: tangents to a circle

thnks bob.
i was not able to construct the diagram.
nw its easy to do it
thnks again.

friendship is tan 90°.

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## #5 2013-10-25 19:35:32

bob bundy
Registered: 2010-06-20
Posts: 8,354

### Re: tangents to a circle

I had three goes before getting a sensible diagram.

"AB produced at Q" was the clue to getting it correct.  In all my previous attempts I had the points in this order A Q B along the line.

Then I realised the questioner was saying A B Q and so I had to twist A around to the other side of BC.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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## #6 2013-10-26 16:40:02

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

### Re: tangents to a circle

thnx 2 nehushtan too fr giving the solution.
@bob i was confused as everytime i was gettin' some haphazard diagrams .