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Question

Question 12
Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of triangle PQR. Show that ΔABCΔPQR.

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Solution

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 ΔABCΔPQR
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)
ΔABDΔPQM [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
ΔABCΔPQR [By SAS similarity criterion]

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