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Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that triangle ABC similar triangle PQR.

Q14  Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and
        median PM of another triangle PQR. Show that \Delta ABC \sim \Delta PQR

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\frac{AB}{PQ}=\frac{AC}{PR}=\frac{AD}{PM}              (given)

Produce AD and PM to E and L such that AD=DE  and PM=DE. Now,

join B to E,C to E,Q to L and R to L.

AD and PM are medians of a triangle, therefore

QM=MR  and    BD=DC

AD = DE            (By construction)

PM=ML               (By construction)

So, diagonals of ABEC bisecting each other at D,so ABEC is a parallelogram.

Similarly, PQLR is also a parallelogram.

Therefore, AC=BE ,AB=EC and PR=QL,PQ=LR

\frac{AB}{PQ}=\frac{AC}{PR}=\frac{AD}{PM}               (Given )

\Rightarrow \frac{AB}{PQ}=\frac{BE}{QL}=\frac{2.AD}{2.PM}

\Rightarrow \frac{AB}{PQ}=\frac{BE}{QL}=\frac{AE}{PL}

  \Delta ABE \sim \Delta PQL                (SSS similarity)

\angle BAE=\angle QPL ...................1           (Corresponding angles of similar triangles)

Similarity, \triangle AEC=\triangle PLR

\angle CAE=\angle RPL........................2

Adding  equation 1 and 2,

\angle BAE+\angle CAE=\angle QPL+\angle RPL

\angle CAB=\angle RPQ............................3

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

\frac{AB}{PQ}=\frac{AC}{PR}        ( Given )

\angle CAB=\angle RPQ        ( From above equation 3)

\triangle ABC\sim \triangle PQR         ( SAS similarity)     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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