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If AD and PM are medians of triangles ABC and PQR, respectively where triangles ABC similar triangles PQR, prove that AB by PQ equals AD by PM

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}

 

 

 

 

 

 

 

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