Question

A $△ABC$ is given. If lines are drawn through $AB,B,C$ parallel respectively to the sides $BC,CA$ and $AB$ forming $△PQR$, as shown in the figure, show that $BC=\frac{1}{2}QR$

Answer 1

We know that $AR\parallel BC$ and $AB\parallel RC$

From the figure we know that $ABCR$ is a parallelogram

So we get

$AR=BC$.... (1)

We know that $AQ\parallel BC$ and $QB\parallel AC$

From the figure we know that $AQBC$ is a parallelogram

so we get

$AQ=BC$.......(2)

By adding both the equations

$AR+QA=BC+BC$

We know that $AR+QA=QR$

so we get

$QR=2BC$

dividing 2

$BC=\frac{QR}{2}$

$BC=\frac{1}{2}QR$

therefore it is proved that $BC=\frac{1}{2}QR$