Question

Through the midpoint 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.

Answer 1

Answer:

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