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Is the Midpoint Theorem for Quadrilaterals (Theorem 9) true when the quadrilateral is non-convex? How about when it is self-intersecting? Experiment and check. The pictures may look strange, but the theorem appears to be true. (i) Can you explain why this is so? See the geometric reasoning in the text. (ii) There is one exception: In a very special case the Varignon parallelogram becomes a single segment. When will this happen? (iii) Does your reasoning apply even when ABCD is non-planar?
Yes, Varignon’s theorem holds generally. (i) The proof relies entirely on applying the triangle Midpoint Theorem to triangles formed by diagonals AC and BD; midpoints always form pairs of opposite sides parallel to and half the length of each diagonal. (ii) The Varignon parallelogram flattens into aRead more
Yes, Varignon’s theorem holds generally. (i) The proof relies entirely on applying the triangle Midpoint Theorem to triangles formed by diagonals AC and BD; midpoints always form pairs of opposite sides parallel to and half the length of each diagonal. (ii) The Varignon parallelogram flattens into a single straight line segment when the two diagonals AC and BD are parallel lines (in self-intersecting figures) or when the four vertices are collinear. (iii) Yes, because each set of three vertices defines a flat triangle where the Midpoint Theorem remains valid in 3D space.
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/
See lessLet 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.
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 pointsRead more
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/
See less(i) Suppose ABCD is a parallelogram and M, N are midpoints of AB and CD respectively. Show that segments DM and BN trisect segment AC. (ii) Use part (i) to find a procedure to trisect any given segment PQ. Find two other ways to trisect PQ, one using Exercise 11 above and a third using the Centroid Theorem.
(i) Since MB = DN and MB ∥ DN, MBND is a parallelogram, making MD ∥ BN. Let DM and BN intersect AC at X and Y. In △ABY, MX ∥ BY through midpoint M gives AX = XY. In △CDX, NY ∥ DX through midpoint N gives XY = YC, proving trisection: AX = XY = YC. (ii) 1. Complete a parallelogram on diagonal PQ. 2. IRead more
(i) Since MB = DN and MB ∥ DN, MBND is a parallelogram, making MD ∥ BN. Let DM and BN intersect AC at X and Y. In △ABY, MX ∥ BY through midpoint M gives AX = XY. In △CDX, NY ∥ DX through midpoint N gives XY = YC, proving trisection: AX = XY = YC. (ii) 1. Complete a parallelogram on diagonal PQ. 2. Intercept three equal segments using parallel lines. 3. Construct a triangle with median PQ; the centroid trisects it in a 2:1 ratio.
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/
See lessComplete the following proof of the Midpoint Theorem. Extend segment PQ beyond Q until point S. By how much should we extend PQ? It would be good to be able to prove that △APQ ≅ △CSQ. (Why?) Use this as a guide to specify the location of S and then complete this proof. (ii) Give a similar proof of the converse of the Midpoint Theorem (Theorem 7). Start by extending PQ up to a suitable point S.
(i) Extend PQ to S such that QS = PQ. Since AQ = CQ and vertical angles match, △APQ ≅ △CSQ by SAS. This gives CS = AP = PB and alternate interior angles show CS ∥ PB. Hence PBCS is a parallelogram. Therefore PQ ∥ BC and PQ = PS/2 = BC/2. (ii) Through C draw a line parallel to AB meeting extended PQRead more
(i) Extend PQ to S such that QS = PQ. Since AQ = CQ and vertical angles match, △APQ ≅ △CSQ by SAS. This gives CS = AP = PB and alternate interior angles show CS ∥ PB. Hence PBCS is a parallelogram. Therefore PQ ∥ BC and PQ = PS/2 = BC/2. (ii) Through C draw a line parallel to AB meeting extended PQ at S, forming parallelogram PBCS with CS = PB = AP. Then △APQ ≅ △CSQ by AAS, which proves AQ = QC.
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/
See lessReview all the properties of a rhombus/rectangle/square that you proved in Grade 8. Formulate a converse of each. Decide if the converse is true. There are many possibilities here!
Rectangle: Property: diagonals are equal. Converse: If diagonals of a parallelogram are equal, it is a rectangle (True); for general quadrilaterals it is False. • Rhombus: Property: diagonals bisect perpendicularly. Converse: If a quadrilateral's diagonals bisect each other perpendicularly, it is aRead more
Rectangle: Property: diagonals are equal. Converse: If diagonals of a parallelogram are equal, it is a rectangle (True); for general quadrilaterals it is False. • Rhombus: Property: diagonals bisect perpendicularly. Converse: If a quadrilateral’s diagonals bisect each other perpendicularly, it is a rhombus (True). • Square: Property: diagonals are equal and perpendicular bisectors. Converse: A quadrilateral with equal diagonals that perpendicularly bisect each other is a square (True).
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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