Question

In the given figure, PQRS and PXYZ are two parallelograms of equal area. Prove that SX || YR.

Answer 1

We are given:

• Two parallelograms PQRS and PXYZ of equal area
• We are asked to prove: SX || YR

Step 1: Recall properties of parallelograms

1. In a parallelogram, opposite sides are parallel.

• In PQRS: PS || QR and PQ || SR
• In PXYZ: PX || YZ and PY || XZ

2. Equal area of parallelograms implies that the height corresponding to a base is the same if the base is the same or bases are proportional to heights otherwise.

Step 2: Draw diagonals or connecting lines

Let us join points SX and YR (as in the figure). We aim to prove these lines are parallel.

• Consider triangles formed by these diagonals and vertices.
• In particular, notice the common base principle:
  • Both parallelograms share a vertex P.
  • Areas of parallelograms: Area(PQRS) = Area(PXYZ)
• Using triangle decomposition, we can write:
  Area(PQRS) = Area(ΔPSR) + Area(ΔPQR) 
  Area(PXYZ) = Area(ΔPXY) + Area(ΔPXZ)
• From the hint in previous problem, if we join XQ and use the fact that triangles with equal base and height have equal area, then the line connecting opposite vertices (SX and YR) must be parallel to preserve equality of areas.

Step 3: Use the triangle on same base, between same parallels property

• The property: Triangles on the same base and between the same parallels have equal areas.
• Since areas of parallelograms are equal, the triangles formed with bases SX and YR are at the same height.
• This is only possible if SX || YR.