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 → Types of Quadrilaterals ↓


Rectangle


Understanding geometry is like solving a never-ending puzzle. In this fascinating puzzle, a key piece is the "rectangle". Rectangles are a fundamental type of quadrilateral, meaning they are four-sided polygons. Studying rectangles helps us understand both the properties of shapes and the logic of mathematical rules.

What is a rectangle?

A rectangle is a special type of quadrilateral in which each opposite side is equal, and each angle is a right angle (90 degrees). The word "rectangle" comes from the Latin words "rectus," meaning "right," and "angulus," meaning "angle."

rectangle

In this diagram, you can see a rectangle with sides ABCD. The features that make it a rectangle are obvious:

  • Opposite sides are parallel and equal: AB = CD, AD = BC.
  • Each angle is 90 degrees, i.e. all are equal.

Properties of rectangles

Let's explore the defining characteristics or properties of rectangles that make them unique and useful in geometry:

  • Opposite sides are equal and parallel: In a rectangle, each pair of opposite sides is equal in length. Mathematically, if you have rectangle ABCD, this means: 
    AB = CD
    AD = BC
  • All angles are right angles: Every angle in a rectangle is 90 degrees. Thus, in rectangle ABCD: 
    ∠A = ∠B = ∠C = ∠D = 90°
  • Diagonals bisect each other: In a rectangle, the diagonals (a line drawn from one corner to the opposite corner) are of equal length and they bisect each other, dividing each into two equal parts. So, for rectangle ABCD: 
    AC = BD
    AO = OC
    BO = OD
    where O is the point of intersection of the diagonals.
  • Symmetry: Rectangles are symmetrical. They have two lines of symmetry: lines passing through the midpoints of opposite sides.

Formulas involving rectangles

When working with rectangles, various formulas become useful, especially when solving problems in geometry.

Perimeter of a rectangle

The perimeter of a rectangle is the total distance around the rectangle. It is calculated by adding the lengths of all four sides. Since opposite sides are equal, the formula is: 

Perimeter = 2 * (Length + Width)
For example, if the length of a rectangle is 8 units and the width is 5 units: 
Perimeter = 2 * (8 + 5) = 26 units

Area of a rectangle

The area of a rectangle is the amount of space contained within its sides. It is determined by multiplying the length by the width: 

Area = Length * Width
Continuing our previous example: 
Area = 8 * 5 = 40 square units

Diagonal of a rectangle

The length of the diagonal in a rectangle can be found using the Pythagorean Theorem, which relates rectangular lengths: 

Diagonal = √(Length² + Width²)
For our rectangle: 
Diagonal = √(8² + 5²) = √(64 + 25) = √89

Textual examples and applications

Understanding rectangles isn't just limited to solving geometry problems. It also extends to real-world applications.

Example 1: Room layout

Imagine you are designing a room. You want to paint it, and you know it is rectangular. You measure the length 6 meters and the width 4 meters. To calculate how much paint you need, first you calculate the perimeter for placing the baseboard trims: 

Perimeter = 2 * (6 + 4) = 20 meters
And if each meter of wall needs a certain amount of paint, knowing the area helps determine how many liters will be needed: 
Area = 6 * 4 = 24 square meters

Example 2: Field design

Consider a playing field that is rectangular in shape. To place a gate at a diagonal corner, you need the diagonal distance. If the field is 100 m by 60 m: 

Diagonal = √(100² + 60²) = √(10000 + 3600) = √13600 = 116.57 m

Visualization of rectangles

Spatial thinking can be enhanced by visualizing rectangles and their properties. Below, try drawing some rectangles of varying sizes using graph paper or coordinate geometry:

Here, the blue rectangle is defined by its length and width on the plane. The red lines represent the diagonals that intersect at the center, confirming that they are equal.

Conclusion

Understanding rectangles opens up the world of geometry as they are not only simple to understand but also practical. With properties such as equal diagonals, symmetry, and simple formulas for perimeter and area, rectangles serve as a solid foundation for more complex geometric concepts.

Continue exploring rectangles by observing them in nature, art, engineering or around your home, as they are all-round useful and fascinating pieces of the mathematical puzzle.


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Rectangle

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