Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Geometry ↓


Understanding Quadrilaterals


Introduction to quadrilaterals

Quadrilaterals are a fascinating and essential concept in geometry, forming the basis of many figures and shapes. A quadrilateral is a polygon with four edges (or sides) and four vertices or corners. The word "quadrilateral" is derived from two Latin words: "quadri" meaning "four", and "latus" meaning "side". This is an important structural understanding in geometry and helps to build further knowledge about polygons.

Basic properties of quadrilaterals

All quadrilaterals have some basic properties. Knowing these properties will help us identify and work with different types of quadrilaterals. Basic properties include:

  • He has four arms.
  • They have four vertices (corners).
  • The sum of their interior angles is always 360 degrees. This is the same in all quadrilaterals. 
    Angle sum property: 360°

Consider the following simple quadrilateral represented visually:

A B C D

Types of quadrilaterals

Quadrilaterals are of different types depending on the lengths of the sides and angles:

1. Parallelogram

The opposite sides of a parallelogram are equal and parallel. In a parallelogram, the opposite angles are also equal. An important property of a parallelogram is that the sum of the adjacent angles is 180 degrees.

A B C D

2. Rectangle

A rectangle is a type of parallelogram, but it has an additional constraint. All interior angles of a rectangle are right angles (90 degrees). Its opposite sides are parallel and of equal length.

Formula for perimeter of rectangle:

Perimeter = 2 * (Length + Breadth)
A B D C

3. Square

A square is a special type of rectangle in which all four sides are equal in length and all four angles are right angles. Since it is a rectangle, the properties of a rectangle apply to it. However, since all sides are equal, it also becomes a rhombus (which we will discuss later).

Formula for area of square:

Area = Side × Side
A B D C

4. Rhombus

A rhombus is like a parallelogram, but all its sides are of equal length. Opposite sides are parallel, and opposite angles are equal. It does not require right angles.

Formula for perimeter of rhombus:

Perimeter = 4 × Side
A B C D

5. Trapezoid (or trapezoid)

A trapezoid is a quadrilateral that has at least one pair of parallel sides. In different fields, it may be called a 'trapezium'. Its main feature is the pair of parallel sides. Its angles and other sides can be of any length.

Consider the following trapezoid:

A B C D

Properties and theorems related to quadrilaterals

Understanding and working with quadrilaterals also involves understanding the various theorems and properties associated with these shapes.

Diagonal properties

Diagonal is a line segment joining two non-adjacent vertices of a polygon. In a quadrilateral, we have two diagonals. These diagonals have different properties depending on the type of quadrilateral.

  • In a rectangle and square the diagonals are equal and bisect each other.
  • In a parallelogram and rhombus the diagonals bisect each other but they are not necessarily equal.

Let's imagine a diagonal in a rectangle:

A B D C

Angle properties

The properties of angles in a quadrilateral can be of great help in solving geometry problems. As mentioned, a basic property is that the sum of the interior angles is 360 degrees. Depending on the additional information given about the quadrilateral, different angle properties may apply.

  • In a rectangle and square all angles are 90 degrees.
  • In a parallelogram the opposite angles are equal.
  • In a rhombus the opposite angles are equal and the adjacent angles are supplementary.

Applications of quadrilaterals

Quadrilaterals are applied in a variety of real-world contexts. Understanding how to work with quadrilaterals is important for fields that include architecture, engineering, computer graphics, and more.

Architecture and engineering

In architecture, various types of quadrilaterals form the basis of many design structures and roofs. Understanding the properties of these shapes helps in designing stable and aesthetically pleasing structures.

Computer graphics

In the field of computer graphics, quadrilaterals are used in mesh modeling. Most 3D models are made up of thousands of small rectangles and other quadrilaterals that are added together to form the surface of the model.

Conclusion

In conclusion, quadrilaterals form an important foundation in geometry. Their properties and types are diverse, providing a platform for understanding more complex geometric concepts. The ability to identify and work with different quadrilaterals has practical applications in various professional fields. Understanding their properties and learning to identify the correct type of quadrilateral when solving problems are important skills for students in mathematics.


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Understanding Quadrilaterals

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