1.

SidesAB and BC and median AD of a triangle ABC are respectively proportional tosides PQ and QR and median PM of `DeltaP Q R`. Show that `DeltaA B C DeltaP Q R`.

Answer» We have,
`(AB)/(PQ)=(BC)/(QR)=(AD)/(PM)`
`rArr (AB)/(PQ)=(AD)/(PM)=(BC)/(QR)=((1)/(2) BC)/((1)/(2)QR)=(BD)/(QM)`
In `Delta ABD and Delta PQM`, We have
`(AB)/(PQ)=(AD)/(PM)=(BD)/(QM)` [ from(i)]
`:. Delta ABD~ Delta PQM` [ by SSS- similarity]
And so, `angleB= angle Q`. [ crosses. angles of similar triangels are equal]
Now, in `Delta ABC and Delta PQR` , we have
`angle B= angle Q` [ proved above]
`and (AB)/(PQ)=(BD)/(QM)` [from (i)]
`:. Delta ABC~ Delta PQR` [ by SAS- similarity]


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