Q6 of 47 Page 7

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

Given: PQ=QR


To prove: LN=LM


Proof:



Here, we can observe that PQR is an isosceles triangle


PQ = QR


And, ∠QPR = ∠QRP (i)


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


PL = LQ =


QM = MR =


And, PQ = QR


PL = LQ = QM = MR = = (ii)


Now, in ∆LPN and ∆MRN


LP = MR (From ii)


∠LPN = ∠MRN (From i)


PN = NR (N is the mid-point of PR)


Hence, By SAS congruency theorem,


∆LPN ≅ ∆MRN


, LN = MN (By c.p.c.t)


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