Through the mid point X of the side RS of a parallelogram PQRS, the line QX is drawn intersecting PR at N and PS produced at E. Prove that EN=2QN.
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Answers
Given : Through the mid point X of the side RS of a parallelogram PQRS, the line QX is drawn intersecting PR at N and PS produced at E.
To Find : Prove that EN=2QN.
Solution:
PQ || RS and PQ = RS ( opposite sides of parallelogram)
=> XS || PQ and XS = PQ/2 as XS = SR/2 ( X being mid point )
Using converse of line joining the mid-point of two sides of a triangle is equal to half the length of the third side
PS = SE
PS = RQ
=> PE = PS + SE = RQ + RQ = 2RQ
=> PE/RQ = 2
in Δ PNE & Δ RNQ
∠PNE = ∠RNQ (Vertically opposite angles )
∠NPE = ∠NRQ ( alternate angle )
∠PEN = ∠RQN ( alternate angle )
=> Δ PNE ≈ Δ RNQ (AAA)
EN/ QN = PE/RQ
=> EN/QN = 2
=> EN = 2 QN
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Hence proved ,EN = 2QN
Hope it helps you .
