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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 (see Fig. 6.41). Show that triangle ABC similar triangle PQR.

Q12  Sides AB and BC and median AD of a triangle ABC are respectively proportional
         to sides PQ and QR and median PM of \Delta PQR (see Fig. 6.41). Show that
         \Delta ABC \sim \Delta PQR

             

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AD and PM are medians of triangles. So,

BD=\frac{BC}{2}\, and\, QM=\frac{QR}{2}

Given : 

           \frac{AB}{PQ}=\frac{BC}{QR}=\frac{AD}{PM}

\Rightarrow \frac{AB}{PQ}=\frac{\frac{1}{2}BC}{\frac{1}{2}QR}=\frac{AD}{PM}

\Rightarrow \frac{AB}{PQ}=\frac{BD}{QM}=\frac{AD}{PM}

In \triangle ABD\, and\, \triangle PQM,

\frac{AB}{PQ}=\frac{BD}{QM}=\frac{AD}{PM}

\therefore \triangle ABD\sim \triangle PQM,         (SSS similarity)

\Rightarrow \angle ABD=\angle PQM      ( Corresponding angles of similar triangles )

 

In \triangle ABC\, and\, \triangle PQR,

\Rightarrow \angle ABD=\angle PQM    (proved above)

\frac{AB}{PQ}=\frac{BC}{QR}

Therefore,\Delta ABC \sim \Delta PQR.   ( SAS similarity)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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