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प्रश्न
PQRS is a parallelogram. M and N are the mid-points of the adjacent sides QR and RS. O is the mid-point of the diagonal PR. Prove that MONR is a rectangle and MN is half of PR.
उत्तर
In ΔSRQ,
N and O are the mid-points of SR and PR respectively.
Therefore, ON || QR and ON = `(1)/(2)"QR"` i.e. ON = MR ........(i)
PR = SQ ...(diagonals of rectangle are equal and bisect each other)
⇒ O is mid-point of SQ
In ΔRQS,
M and O are the mid-points of QR and SQ respectively.
Therefore, OM || SR and OM = `(1)/(2)"QR"` i.e. OM = SQ ........(ii)
∠MRN = ∠QRS = 90° ...........(iii) (PQRS is a rectangle)
From (i) , (ii) and (iii)
Therefore, quadrilateral MONR has two opposite pairs of sides equal and parallel and an interior angle as right angle, so it is a rectangle.
In ΔSQR,
M and N are the mid-points of QR and SR respectively.
Therefore, MN || SQ and MN = `(1)/(2)"SQ"`
But SQ = PR
⇒ MN = `(1)/(2)"PR"`.
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