New NCERT Ganita Manjari · Class 9 Maths · Part II · Chapter 12: Quadrilaterals
This chapter defines a quadrilateral exactly and explains convex and non-convex 4-gons. It proves the properties of parallelograms and the tests for a parallelogram, the Midpoint Theorem, the Centroid Theorem and Varignon's Theorem. It ends by showing that any quadrilateral can tile the plane.
From the Class 9 Maths Formula Sheet, made as per the new NCERT book Ganita Manjari · Pages 58–62 of the book.
Key points
- Tiling means covering the whole plane with copies of a shape, with no gaps and no overlaps.
- A quadrilateral ABCD is formed by four distinct points in a plane, no three collinear, joined by four segments that meet only at the vertices.
- The angle sum of any quadrilateral is 360°, for convex and non-convex ones.
- In a convex quadrilateral every internal angle is less than 180° and the diagonals intersect; a non-convex one has an angle more than 180°.
- In a parallelogram, opposite sides are equal, opposite angles are equal and the diagonals bisect each other (Theorem 1).
- The converses are all true and give tests for a parallelogram, including one pair of opposite sides equal and parallel.
- Midpoint Theorem: the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length; its converse is also true.
- Joining the midpoints of the three sides divides a triangle into 4 congruent triangles.
- The three medians of a triangle meet at the centroid G, which divides each median in the ratio 2 : 1.
- Varignon's theorem: the midpoints of the sides of any quadrilateral form a parallelogram.
- Any quadrilateral can tile the plane, but a regular pentagon cannot.
- A triangle with sides a ≤ b ≤ c exists exactly when a + b > c.
Formulas
| Angle sum of a quadrilateral | ∠A + ∠B + ∠C + ∠D = 360° |
| Midpoint Theorem | PQ ∥ BC and PQ = ½ BC P, Q midpoints of AB, AC |
| Midpoint theorem for a trapezium | EF = ½(AB + CD) AB ∥ DC, E and F midpoints of AD and BC |
| Centroid Theorem | AG : GR = BG : GQ = CG : GP = 2 : 1 AG = ⅔ AR, GR = ⅓ AR |
| Area of the midpoint triangle | Area of △PQR = ¼ area of △ABC |
| Perimeter of the Varignon parallelogram | = AC + BD |
| Sum of interior angles of an n-gon | (n − 2) × 180° |
| Number of diagonals of an n-gon | n(n − 3)/2 |
| Each angle of a regular n-gon | (n − 2) × 180° ÷ n |
Diagrams to learn
- Figures with 4 points that are and are not quadrilaterals (NOPE, SILY, DART, CUTS, OPENS)
- Table of parts of a quadrilateral
- Convex and non-convex quadrilaterals
- Venn diagram of the family of 4-gons
- Parallelogram with diagonals bisecting each other
- Midpoint Theorem construction
- Four congruent triangles from midpoints
- Medians and centroid of a triangle
- Varignon parallelogram
- Tiling the plane with any quadrilateral
- Polygon formula table
- A rhombus with perpendicular diagonals is not necessarily a square — every rhombus has perpendicular diagonals.
- A quadrilateral with equal diagonals need not be a rectangle, e.g. an isosceles trapezium.
- A regular pentagon cannot tile the plane: each angle is 108° and 360° ÷ 108° is not a whole number.
- To prove a converse, try running the original proof backwards, but justify every step again.
A real page from the book
Get the full chapter in the book
This page is a summary. The Class 9 Maths Formula Sheet has the complete chapter in colour, with every diagram and worked example.
Frequently asked questions
What are the properties of a parallelogram?
In a parallelogram, opposite sides are equal, opposite angles are equal, and the diagonals bisect each other.
What is the Midpoint Theorem in Class 9?
The segment joining the midpoints of two sides of a triangle is parallel to the third side and is half of it: PQ ∥ BC and PQ = ½ BC.
In what ratio does the centroid divide a median?
The centroid G divides each median in the ratio 2 : 1, with the longer part towards the vertex.
← Part II · Chapter 11: The World of AlgorithmsPart II · Chapter 13: Two Variables, One Line →
