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Question

Sides AB,AC and median AD of a triangle ABC are respectively proportional to sides PQ,PR and median PM of another triangle PQR. Show that ABCPQR.

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Solution

given:ABPQ=ACPR=ADPM=k(let)

AB=kPQ(i)

AC=kPR(ii)
length of median

AD=122(AB2+AC2)BC2

PM=122(PQ2+PR2)QR2

using(i)&(ii)

AD=122k2(PQ2+PR2)BC2

ADPM=k

AD2PM2=k2

2k2(PQ2+PR2)BC22(PQ2+PR2)QR2=k2

2k2(PQ2+PR2)BC2=2k2(PQ2+PR2)QR2(k)2

BC2=QR2(k)2

orBCQR=k

hence,ABPQ=ACPR=BCQR

ABCPQRbyS.S.Ssimilarity

1000924_1052191_ans_d38bbbb948e84b9f80596dd6b8d86e95.png

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