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PQRS is a ‖parallelogram . M is a point on PS such that PM = 7 PS and N is apoint on QR such that RN = 7 QR. Prove that the quadrilateral PNRM is a ‖gm.

Question

PQRS is a ‖parallelogram . M is a point on PS such that PM = 7 PS and N is apoint on QR such that RN = 7 QR. Prove that the quadrilateral PNRM is a ‖gm.

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Solution

To prove that the quadrilateral PNRM is a parallelogram, we need to show that its opposite sides are parallel.

  1. Given that PQRS is a parallelogram, we know that its opposite sides are parallel. Therefore, PQ || SR and PS || QR.

  2. Let's consider the line segment PM. We are given that PM = 7PS. This means that PM is 7 times longer than PS.

  3. Similarly, let's consider the line segment RN. We are given that RN = 7QR. This means that RN is 7 times longer than QR.

  4. Now, let's draw a line segment from P to R. Since PQ || SR, the line segment PR is parallel to QR.

  5. Since PM is 7 times longer than PS and RN is 7 times longer than QR, it follows that PR is also 7 times longer than PS.

  6. Therefore, we can conclude that PR is parallel to PS.

  7. Similarly, we can draw a line segment from N to M. Since PS || QR, the line segment NM is parallel to PS.

  8. Since RN is 7 times longer than QR and PM is 7 times longer than PS, it follows that NM is also 7 times longer than PS.

  9. Therefore, we can conclude that NM is parallel to PS.

  10. We have shown that PR is parallel to PS and NM is parallel to PS. This means that PNRM is a quadrilateral with opposite sides that are parallel.

  11. Hence, we have proved that the quadrilateral PNRM is a parallelogram.

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