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In a ΔPQR, if PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP respectively. Prove that LN = MN.

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Given that in  ΔPQR, PQ = QR and, L M and N are mid-points of PQ, QR and RP respectively

We have to prove  LN = MN

Join L and M, M and N, N and L

We have

PL = LQ, QM = MR and RN = NP

[ ∴ L, M and N are mid-points of PQ, QR and RP respectively]

And also

PQ = QR= PL = LQ = QM = MR=PQ/2=QR/2  ....(1) Using mid-point theorem, we have

MN || PQ and MN=(1/2) x PQ

⇒MN=PL=LQ    .......(2)

Similarly, we have

LN QR and LN=(1/2) x QR

⇒ LM=QM MR  .........(3)

From equation (1), (2) and (3), we have

PL= LQ= QM= MR= MN= LN

LN=MN

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