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. Show that :

(i) ∆ABM ≅ ∆PQN 

(ii) ∆ABC ≅ ∆PQR.

Answer 1

Data : 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. 

To Prove: 

(i) ∆ABM ≅ ∆PQN 

(ii) ∆ABC ≅ ∆PQR. 

Proof: 

(i) In ∆ABC, AM is the median drawn to BC. 

∴ BM = \(\frac{1}{2}\)BC 

Similarly, in ∆PQR, 

QN = \(\frac{1}{2}\)QR 

But, BC = QR 

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

∴ BM = QN 

In ∆ABM and ∆PQN, 

AB = PQ (data) 

BM = QN (data) 

AM = PN (proved) 

∴ ∆ABM ≅ ∆PQN (SSS postulate) 

(ii) In ∆ABC and ∆PQR, 

AB = PQ (data) 

∠ABC = ∠PQR (proved) 

BC = QR (data) 

∴ ∆ABC ≅ ∆PQR (SSS postulate)