Question

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

Solution

## We know that $$AR\parallel BC$$ and $$AB\parallel RC$$From the figure we know that $$ABCR$$ is a parallelogramSo we get$$AR=BC$$.... (1)We know that $$AQ\parallel BC$$ and $$QB\parallel AC$$From the figure we know  that $$AQBC$$ is a parallelogramso 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$$BC=\dfrac{QR}{2}$$$$BC=\dfrac{1}{2} QR$$therefore it is proved that $$BC=\dfrac{1}{2}QR$$MathematicsRS AgarwalStandard IX

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