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Q16   If AD and PM are medians of triangles ABC and PQR, respectively where
       \Delta AB C \sim \Delta PQR , prove that \frac{AB}{PQ} = \frac{AD }{PM}
 

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\Delta AB C \sim \Delta PQR     ( Given )

\frac{AB}{PQ}=\frac{AC}{PR}=\frac{BC}{QR}     ............... ....1( corresponding sides of similar triangles )

\angle A=\angle P,\angle B=\angle Q,\angle C=\angle R....................................2

AD and PM are medians of triangle.So,

BD=\frac{BC}{2}\, and\, QM=\frac{QR}{2}   ..........................................3

From equation 1 and 3, we have

  \frac{AB}{PQ}=\frac{BD}{QM}...................................................................4

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

 \angle B=\angle Q        (From equation 2)

\frac{AB}{PQ}=\frac{BD}{QM}        (From equation 4)

\triangle ABD\, \sim \, \triangle PQM,        (SAS similarity)

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

 

 

 

 

 

 

 

Posted by

seema garhwal

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