the 11. In the adjoining figure, ABC is a triangle. Through A, B and C lines are drawn parallel to BC, CA and AB respectively, which forms a triangle PQR. Show that 2(AB + BC + CA) = PQ + QR + RP. rer.
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From the figure, it is given that, Through A,B and C lines are drawn parallel to BC,CA and AB respectively.
We have to show that 2(AB+BC+CA)=PQ+QR+RP
∵ AB∥RC and AR∥CB
Therefore, ABCR is a parallelogram.
⟹AB=CR and CB=AR (opposite sides of a parallelogram are equal) -------(1)
Similarly, ABPC is a parallelogram (∵AB∥CP and PB∥CA)
⟹AB=PC and AC=PB (opposite sides of a parallelogram are equal) -------(2)
Similarly, ACBQ is a parallelogram (∵BC∥AQ and AC∥BQ)
⟹AC=BQ and BC=AQ (opposite sides of a parallelogram are equal) --------(3)
By adding the equations (1), (2) and (3) we get,
AB+AB+BC+BC+AC+AC=PB+PC+CR+AR+BQ+AQ
2AB+2BC+2AC=PQ+QR+RP
By taking common we get,
2(AB+BC+AC)=PQ+QR+RP
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