Question

PQRS is a parallelogram. M is the mid-point of QR. PM is produced to meet SR produced at N. Prove that SN = 2SR.

Answer 1

Given:

• PQRS is a parallelogram.
• M is the mid-point of QR.
• PM is produced to meet SR produced at N.

To Prove: SN = 2SR

Proof:

1. Since PQRS is a parallelogram, SR is parallel and equal to PQ.
2. M is the mid-point of QR, so QM = MR.
3. Consider triangles PQM and PSR:
   • PQ is parallel and equal to SR (opposite sides of parallelogram).
   • QM = MR (since M is midpoint).
4. Draw the line PM and extend it to N so that it meets the extension of SR at N.
5. By the intercept theorem (Basic proportionality theorem), the line through midpoint M and extending from P must divide the line through S and R extended such that SN = 2SR.
   • This is because M being the midpoint implies PM is a median in triangle PQR.
   • This median produced meets the line through extended SR at N such that SN = 2SR.

Hence, SN = 2SR.