Question

In a 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.

Answer 1

In △PQR, PQ = QR and L, M, N are midpoints of the sides PQ, QP and RP respectively (Given) 

To prove : LN = MN 

As two sides of the triangle are equal, so △ PQR is an isosceles triangle 

PQ = QR and ∠QPR = ∠QRP ……. (i) 

Also, L and M are midpoints of PQ and QR respectively 

PL = LQ = QM = MR = QR/2 

Now, consider Δ LPN and Δ MRN, 

LP = MR 

∠LPN = ∠MRN [From (i)] 

∠QPR = ∠LPN and ∠QRP = ∠MRN 

PN = NR [N is midpoint of PR] 

By SAS congruence criterion, 

Δ LPN ≃ Δ MRN 

We know, corresponding parts of congruent triangles are equal. 

So LN = MN

Proved.