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Piyush365
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Asked: 2026-10-05T07:29:28+00:00 In:

Let P, Q, R, S be four points on sides AB, BC, CD and DA respectively of a quadrilateral ABCD. Suppose PQRS is a parallelogram. Must P, Q, R and S be midpoints of the respective sides? This can be considered a possible converse question to Theorem 9.

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No, they do not have to be midpoints. In a parallelogram ABCD, choosing points dividing opposite sides in equal ratios (like AP = CR = 1/3 AB and BQ = DS = 1/3 BC) creates an inscribed parallelogram that is not formed by midpoints.

Class 9 Ganita Manjari Part 2 chapter 12 Quadrilaterals solutions
Class 9 Ganita Manjari Part 2 book Solutions

chapter 12 quadrilaterals solutionsconsidered a possible converse question to theorem 9download class 9 maths ganita manjari part 2 book and solutionsncert solutions – ganita manjari part ii quadrilateralssuppose pqrs is a parallelogram
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    1. No, P, Q, R and S need not be the midpoints of the sides. For example, take any parallelogram ABCD and choose points P, Q, R, S such that AP = CR = 1/3 AB and BQ = DS = 1/3 BC. Then △APS ≅ △CRQ and △PBQ ≅ △RDS by SAS congruence, which forces PQRS to be a parallelogram even though none of the points are midpoints.

       

      For more NCERT Solutions of Class 9 Maths Ganita Manjari Part 2 Chapter 12 Quadrilaterals Question Answer (2026-27)

      https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-12/

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