Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Quadrilaterals ↓


Properties of Quadrilaterals


A quadrilateral is a polygon with four edges (or sides) and four corners or vertices. Its name comes from "quad" meaning four, and "lateral" meaning sides. In this section, we will explore the different types of quadrilaterals and their properties.

Quadrilaterals can be divided into different categories based on their sides and angles. The main types of quadrilaterals are:

  1. Quadrilateral
  2. Rectangle
  3. Social class
  4. Rhombus
  5. Trapezoid (or trapezium in some countries)
  6. Kite

Types of quadrilaterals and their properties

1. Parallelogram

A parallelogram is a quadrilateral whose opposite sides are parallel and equal in length. Opposite angles are equal, and the sum of consecutive angles is 180 degrees.

ABCD

Properties of parallelogram:

  • The opposite sides are parallel and of equal length.
  • Opposite angles are equal.
  • The diagonals bisect each other.
  • The sum of the interior angles is 360 degrees.
Example: AB = CD, AD = BC Angle A = Angle C, Angle B = Angle D

2. Rectangle

A rectangle is a type of parallelogram, all of whose angles are right angles (90 degrees).

ABCD

Properties of rectangle:

  • The opposite sides are equal and parallel.
  • All angles are 90 degrees.
  • The diagonals are equal and bisect each other.
Example: AB = CD, AD = BC All angles = 90 degrees Diagonals: AC = BD

3. Square

A square is a special type of rectangle in which all sides are of equal length and all angles are right angles.

ABCD

Properties of the square:

  • All sides are equal and parallel.
  • All angles are 90 degrees.
  • The diagonals are equal, bisect each other, and are perpendicular to each other.
Example: All sides = AB = BC = CD = DA All angles = 90 degrees Diagonals: AC = BD and AC ⊥ BD

4. Rhombus

A rhombus is a quadrilateral whose all sides are of equal length. Opposite sides are parallel, and opposite angles are equal.

ABCD

Properties of rhombus:

  • All sides are equal.
  • The opposite sides are parallel.
  • Opposite angles are equal.
  • The diagonals bisect each other at right angles (90 degrees).
Example: All sides = AB = BC = CD = DA Opposite angles: Angle A = Angle C, Angle B = Angle D Diagonals AC ⊥ BD

5. Trapezoid

A trapezoid is a quadrilateral in which at least one pair of opposite sides is parallel.

ABCD

Properties of trapezoid:

  • One pair of opposite sides is parallel (these are called the 'bases').
  • The non-parallel sides are called 'legs'.
  • Angles on the same leg are complementary (the sum is 180 degrees).
Example: AB ∥ CD Angles: Angle A + Angle D = 180 degrees, Angle B + Angle C = 180 degrees

6. Kite

A kite is a quadrilateral with two different pairs of equal adjacent sides.

ABCD

Properties of kite:

  • Two pairs of adjacent sides are equal.
  • A pair of opposite angles are equal, which are the angles between unequal sides.
  • Diagonals bisect each other at right angles.
  • One of the diagonals is the line of symmetry.
Example: AB = AD, BC = CD Opposite angles: Angle A = Angle C Diagonals AC ⊥ BD

Mathematical properties of quadrilaterals

In addition to the definitions and properties described above, all quadrilaterals have some common mathematical properties:

  • The sum of all the interior angles of any quadrilateral is 360 degrees. It can be found using the formula (n-2) * 180, where n is the number of sides. For a quadrilateral, n = 4.
  • Based on the given sides and angles, you can often calculate the diagonals and area using specific formulas that relate to that quadrilateral.

Calculating the area of quadrilaterals

The area of a quadrilateral can be calculated in several ways, depending on the type of quadrilateral:

  • Rectangle: Area = length * width
  • Square: Area = side * side or side 2
  • Parallelogram: Area = base * height
  • Rhombus: Area = (diagonal 1 * diagonal 2) / 2
  • Trapezium: Area = (base 1 + base 2) * height / 2
  • Kite: Area = (diagonal 1 * diagonal 2) / 2
Example for Trapezium: Let base 1 = a, base 2 = b, and height = h Area = (a + b) * h / 2

Example problem with quadrilaterals

Let's solve a problem to better understand how these properties can be applied:

Problem: Find the area of a kite with diagonals 8 cm and 6 cm.

Solution:

Given: Diagonal 1 = 8 cm Diagonal 2 = 6 cm Using the area formula for a kite: Area = (Diagonal 1 * Diagonal 2) / 2 = (8 * 6) / 2 = 48 / 2 = 24 cm 2

Conclusion

Quadrilaterals, as fundamental geometric shapes, provide the basis for understanding more complex polygons and spatial reasoning in mathematics. Becoming familiar with their properties - such as side lengths, angle measures, parallel lines and diagonals - aids in solving practical and theoretical problems. Whether you are calculating areas, drawing diagrams or solving geometry problems, quadrilaterals provide a rich field of study in mathematics.

As you continue to explore these shapes, remember that mastering basic properties and relationships can make solving more complex tasks much easier. This understanding lays the foundation for deepening your mathematical knowledge and enhancing your problem-solving skills.


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Properties of Quadrilaterals

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