Question

Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ∆ PQR (see Fig. 7.40). Show that: 

(i) ∆ BM≅∆ PQN 

(ii) ∆ ABC≅∆ PQR

Answer 1

(i) In ∆ABC, AM is the median to BC.

∠ BM=\(\frac{1}{2} \) BC

∠QN= \(\frac{1}{2} \) QR

However, BC =QR

∠BC=\(\frac{1}{2} \) QR= \(\frac{1}{2} \)

∠BM=QN ….(1)

In ∆ABM and ∆PQN,

In ∆PQR, PN is the median to QR.

AB = PQ (Given) 

BM = QN [From equation (1)] 

AM = PN (Given) 

∠∆ABM ≅ ∠∆PQN (SSS congruence rule) 

∠ABM = ∠PQN (By CPCT) 

∠ABC =∠PQR … (2) 

(ii) In ∆ABC and ∆PQR, 

AB = PQ (Given) 

∠ABC = ∠PQR [From equation (2)] 

BC = QR (Given) 

∠∆ABC ≅ ∠∆PQR (By SAS congruence rule)