In the given figure, PQR is a triangle in which PQ = PR. QM and RN are the medians of the triangle. Prove that

(i) ΔNQR = ΔMRQ

(ii) QM = RN

(iii) ΔPMQ = ΔPNR

ΔPQR is an isosceles triangle. [∵ PQ = PR]

⇒1/2 PQ =1/2 PR

⇒ NQ = MR and PN = PM

(i) In ΔNQR and ΔMRQ

NQ = MR (Half of equal sides)

∠NQR = ∠MRQ (Angles opposite to equal sides)

QR = RQ (Common)

ΔNQR = ΔMRQ (By SAS rule)

(ii) QM = RN (Congruent parts of congruent triangles)

(iii) In ΔPMQ and ΔPNR

PN = PM (Half of equal sides)

PR = PQ (Given)

∠P = ∠P (Common)

ΔPMQ = ΔPNR (By SAS rule)