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)