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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)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Posted by

seema garhwal

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