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Algebra

Cool goIITian

Joined:	10 Aug 2007
Post:	33

1 Jan 2008 09:13:44 IST

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345

series(iit 98 question)

None

in a triangle PQR,sinP,sinQ,sinR are inA.P., then

a) the altitudes are in a.p.

b) the altitudes are in h.p.

c) the medians are in g.p.

d) the medians are in a.p.

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Comments (2)

waterdemon

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Joined: 11 Jun 2007

Posts: 1048

1 Jan 2008 13:02:32 IST

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Let side opposite to angle P be "p"
Let side opposite to angle Q be "q"
Let side opposite to angle R Be "r"

Given to us:
SinP , SinQ , SinR are in A.P
Therefore,
p,q,r will also be in A.p as in a triangle the lenght of the
sides are corresponding to their angles.

Let,
SinP/p = SinQ/q = SinR/r = K .......(1)

Also,we know altitudes are
qrSinP/p , prSinQ/q , pqSinR/r

Now using equation (1) we have:

q/pr.SinQ - p/qr.SinP = q/K.pqr - p/k.pqr = q-p/K.pqr

And Similarly:
r/pq.SinR - q/pr.SinQ = r-q/K.pqr

Now we will satisfy condition of p,q,r to be in AP.
So we put
q-p = r-q
we get p,q,r in A.P

Therefore,
r/pq.SinR - q/pr.SinQ = q/pr.SinQ - p/qr.SinP

Now we get,
2(q/pr.SinQ) = r/pq.SinR + p/qr.SinP

So we have,
p/qr.SinP , q/pr.SinQ , r/pq.SinR in A.P

And

qr.SinP/p , pr.SinQ/q , pq.SinR/r in H.P
Hence answer...
altitudes are in H.P

Proved.

Hope you find it useful.
Cheers!!!!!!!!!@@@!!!!!!!!!!

ankita singh

Cool goIITian

Joined: 10 Aug 2007

Posts: 33

1 Jan 2008 18:47:46 IST

Like 0 people liked this

thnx a lot !!

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