Answer :


Given that,


In Δ PQR


PQ = QR


And,


L, M, and N are the mid points of PQ, QR and RP respectively


To prove: LM = MN


Construction: Join L and M, M and N and N and L


Proof: We have,


PL = LQ, QM = MR and RN = NP


Since, L, M and N are mid points of PQ, QR and RP respectively.


And, also PQ = QR


PL = LQ = QM = MR = = (i)


Using mid-point theorem, we have


MN PQ


And,


MN = PQ = MN = PL = LQ (ii)


Similarly, we have


LN QR


And,


LN = QR = LN = QM = MR (iii)


From equations (i), (ii) and (ii), we have


PL = LQ = QM = MR = MN = LN


Therefore, LN = MN


Hence, proved


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