The side MN of a parallelogram MNOP is produced to Q such that MN = NQ. PQ intersects ON at R. The point R divides ON in the ratio:

Correct — D. Draw MNOP in that vertex order, then extend MN beyond N to Q with NQ = MN.

In parallelogram MNOP the side PO is parallel to MN and equal to it.
Q sits on MN produced with NQ = MN, so NQ is parallel to PO and the same length.

Two segments that are equal, parallel and pointing the same way give O − P = Q − N, which rearranges to O + N = P + Q.
That is exactly the statement that ON and PQ share a midpoint.

So R, where PQ cuts ON, is the midpoint of ON: OR = RN, a ratio of 1 : 1 → option (d)

Why the others are wrong

• (a)
  1 : 2 — 1 : 2 would put R a third of the way down ON. Test it with M(0,0), N(1,0), O(1,1), P(0,1): then Q is (2,0) and PQ meets ON at (1, 0.5), the exact midpoint.
• (b)
  3 : 1 — 3 : 1 would place R three-quarters of the way from O. The identity O + N = P + Q forces the crossing to be the midpoint, so the two parts are equal whatever the parallelogram's shape.
• (c)
  2 : 1 — 2 : 1 is the centroid ratio, which belongs to the medians of a triangle. What governs here is that POQN is itself a parallelogram, and a parallelogram's diagonals bisect each other.

The whole question turns on one test: a quadrilateral with one pair of opposite sides both equal and parallel is a parallelogram.

MN = PO because MNOP is a parallelogram, and NQ = MN by construction, so NQ = PO and the two are parallel. That makes POQN a parallelogram in its own right — PO and QN are the equal, parallel pair — and ON and PQ are its two diagonals.

The diagonals of a parallelogram bisect each other, so their meeting point R is the midpoint of ON — and of PQ as well.

Key facts

• A quadrilateral with one pair of opposite sides equal and parallel is a parallelogram.
• The diagonals of a parallelogram bisect each other.
• In vectors, O − P = Q − N rearranges to O + N = P + Q, which says ON and PQ have the same midpoint.
• The vertex order MNOP means MN and PO are opposite sides, as are NO and MP.

Common traps

• Importing the 2 : 1 centroid ratio, which applies to medians of a triangle rather than to this figure.
• Assuming the answer depends on the parallelogram's shape when it holds for every one.
• Misreading the vertex order MNOP and pairing the wrong sides as opposite.

No directly related past PYQ was found.