Question

In a parallelogram PQRS, M and N are the midpoints of the opposite sides PQ and RS respectively. Prove that
RN and RM trisect QS.

Answer 1

Since M and N are the mid-point of PQ and RS respectively.

∴ PM = `(1)/(2)"PQ" and "RN" = (1)/(2)"RS"`  ....(i)

But PQRS is a parallelogram,
∴ PQ = RS and PQ || RS

⇒ `(1)/(2)"PQ" = (1)/(2)"RS"` and PQ || RS

⇒ PM = RN and PQ || RS
⇒ PMRN is a parallelogram.
⇒ PN || RM
⇒ NY || RX                   ....(ii)
We know that the segment drawn through the mid-point of one side of a triangle and parallel to the other side bisects the third side.
In ΔSRX, N is the mid-point of RS and NY || RX                   ....[From (ii)]
∴ Y is the mid-point of QY
⇒ XY = YS                    ....(iii)
Similarly, in ΔPQY, M is the mid-point of PQ and MX || PY   ....[From (ii)]
⇒ QX = XY                   ....(iv)
From (iii) and (iv), we get
QX = XY = YS
⇒ X and Y trisect QS
⇒ PN and RM trisect QS.