Given,ΔABC and ΔPQR,AB,BC and median AD of ΔABC are proportional to sides PQ, QR and median PM of ΔPQR
i.e., ABPQ=BCQR=ADPM
To Prove that ΔABC∼ΔPQR
Proof
ABPQ=BCQR=ADPM
⇒ABPQ=12BC12QR=ADPM.....(i)
⇒ABPQ=BDQM=ADPM (D is the mid-point of BC. M is the mid-point of QR)
⇒ΔABD∼ΔPQM [SSS similarity criterion]
∴∠ABD=∠PQM [Corresponding angles of two similar triangles are equal]
⇒∠ABC=∠PQR
In ΔABC and ΔPQR
ABPQ=BCQR...(i)
∠ABC=∠PQR...(ii)
From equation (i) and (ii), we get
ΔABC∼ΔPQR [By SAS similarity criterion]