Question

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. 

Answer 1

Given that, in PQR, PQ QRand L,M,N are midpoints of the sides PQ, QP and RP
respectively and given to prove that LN  MN
Here we can observe that PQR is and isosceles triangle
⇒PQ =QR and ∠QPR =∠QRP ……..(1)
And also, L and M are midpoints of PQ and QR respectively 
⇒ `PL=LQ=(PQ)/2,QM=MR=(QR)/2` 

And also, PQ=QR 

⇒ `PL=LQ=QM=MR=(PQ)/2=(QR)/2` .............(2) 

Now, consider ΔLPN  and ,Δ MRN 
LP= MR           [From – (2)]
∠LPN =∠MRN   [From – (1)]
∵∠QPR  and ∠LPN  and ∠ QRP  and ∠MRN are same
PN= NR  [∵N is midpoint of PR]
So, by SAS congruence criterion, we have LPN≅  MRN 
⇒LN =MN
[ ∵Corresponding parts of congruent triangles are equal]