Question

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.

Answer 1

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"`.