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Question
PQRS is a parallelogram. T is the mid-point of RS and M is a point on the diagonal PR such that MR = `(1)/(4)"PR"`. TM is joined and extended to cut QR at N. Prove that QN = RN.
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Solution
Join PR to intersect QS at O
Diagonals of a parallelogram bisect each other.
Therefore, OP = OR
But MR = `(1)/(4)"PR"`
∴ MR = `(1)/(4)(2 xx "QR")`
⇒ MR = `(1)/(2)"OR"`
Hence, M is the mid-point of OR.
In ΔROS, T and M are the mid-points of RS and OR respectively.
Therefore, TM || OS
⇒ TN || QS
Also in ΔRQS, T is the mid-point of RS and TN || QS
Therefore, N is the mid-point of QR and TN = `(1)/(2)"QS"`
⇒ QN = RN.
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