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 ↓


Constructions


Construction in geometry means drawing shapes, angles and lines accurately. In class 8 maths, geometric constructions are usually done using a compass and straightedge (ruler without measurement markings). The essence of these constructions is not just to draw shapes but to understand the properties and relationships of geometric shapes. Let's dive into the world of geometric constructions.

Basic tools for construction

To perform geometric constructions we use the following primary tools:

  • Compass: Used to draw arcs and circles.
  • Straightedge: Used to draw straight lines.

These tools allow us to create precise geometric shapes with accuracy.

Basic construction

The basic constructs we frequently explore include:

  • Constructing the bisector of a given angle.
  • Constructing a perpendicular bisector of a given line segment.
  • Constructing angles of specific measures, such as 60 degrees, 90 degrees, etc.

Constructing a bisector of a given angle

An angle bisector is a line that divides an angle into two equal parts. Here's how you can draw it:

  1. Place the compass at the vertex of the angle (point A). Draw an arc that intersects both sides of the angle (the arc intersects at points B and C).
  2. Without changing the compass width, place the compass at point B and draw an arc within the angle.
  3. Using the same compass width, place the compass at point C and draw another arc within the angle that intersects the first arc. Label the intersection point as D.
  4. Draw a straight line from point A to point D. This line is the angle bisector.
    Angle: ∠BAC
    Arc BC intersects sides AB and AC, arcs from B and C intersect at D.
    Line AD is the bisector.
A B C D

Constructing a perpendicular bisector of a line segment

The perpendicular bisector is a line that is at right angles to a line segment and divides it into two equal parts. To construct it:

  1. Place your compass at one endpoint of the line segment (point P), and set it to just over half the length of the line segment.
  2. Draw arcs above and below the line.
  3. Without changing the compass width, repeat this at the other end point (point Q), creating intersections with the previous arcs. Label these intersection points as R and S.
  4. Draw a line through R and S. This line is the perpendicular bisector.
    Line segment: PQ
    The arcs from P and Q intersect at R and S.
    Line RS is the perpendicular bisector.
P Q S R

Creating specific angles

Drawing a 60 degree angle

A common construction is to draw a 60 degree angle, often used to construct an equilateral triangle:

  1. Draw a straight line AB.
  2. Place your compass at point A and draw an arc that intersects line AB. Name the intersecting point C.
  3. Without changing the width of the compass, place it at point C and draw another arc, keeping the compass at the same width.
  4. Label the new intersection with the arc drawn from A as D.
  5. Draw the line segment AD. ∠BAD is a 60 degree angle.
    Line: AB
    The arcs from A and C form intersection D.
    ∠BAD = 60°
A B C D

Using these simple techniques, many other constructions become possible, such as copying segments, creating angles of different measures, and even constructing tangents to circles.

Properties of constructed shapes

When building, it is important to understand the properties of the shapes being built:

  • Angle Bisector: Creates two congruent angles.
  • Perpendicular bisector: Equal distance from both endpoints of a line segment.
  • 60 degree angle: Part of an equilateral triangle.

Why learn construction work?

Learning construction is essential because:

  • They provide the basis for understanding more complex geometric concepts.
  • They improve spatial reasoning and the ability to visualize different shapes and their properties.
  • They provide a practical approach to learning geometry that complements theoretical knowledge.

Additional creative exercises

In addition to the basics, here are some more challenging exercises you might want to try:

  • To bisect a 60 degree angle to form a 30 degree angle.
  • Construct an equilateral triangle given one side.
  • Constructing parallel lines through a point which does not lie on the line.

These exercises broaden your understanding and enhance your skills in geometric constructions.

Conclusion

Constructions in geometry open up a world where measurement inaccuracies are avoided. Instead of using a ruler for measurement, constructions rely on defined steps and properties that keep figures accurate. Mastering constructions requires practice and patience, and even then, it reveals fascinating relationships within geometry. From simple angle bisectors to complex geometric patterns, every construction has underlying principles to unlock, making geometry not only systematic but also deeply satisfying.


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Constructions

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