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 ↓


Constructions


Mathematical construction is a way of accurately drawing shapes, angles, and lines. While graphs are made using machines or computers, constructions are done using tools such as a compass and straightedge (ruler). The process does not require measurements with a protractor or a ruler with units, but only concepts of geometric principles. Let's explore various construction techniques that are fundamental in the field of mathematics.

Tools used in construction

The tools needed for basic construction are:

  • Compass: An instrument used to draw arcs and circles. It can be adjusted to different sizes.
  • Straight line: A ruler without measurement markings is used to draw straight lines.

Basic construction steps

Before we discuss specific types of construction, let’s look at the basic steps involved in any construction.

  1. Identify the type of construction required, such as a line segment, angle, or geometric shape.
  2. Have your compass and straightedge handy as needed.
  3. Perform the steps sequentially, ensuring that precision is maintained.
  4. Carefully check the accuracy of the completed construction.

Construction of a line segment

A line segment is a part of a line bounded by two distinct end points. Here is how to construct a line segment of a given length:

  1. Place the tip of the compass at one of the end points, say point A, and open it to the desired length of the line segment.
  2. Without changing the width of the compass, draw an arc and mark the intersection point as B.
  3. Use a straight line to connect points A and B.

Example: Let's draw a line segment AB of length 5 cm: Open your compass to 5 cm, place the point at A, and draw an arc with the compass. Mark the intersection as B, and draw a straight line AB.

AB --|-----------------------------------------|-- |----------5 cm------------|
AB --|-----------------------------------------|-- |----------5 cm------------|

Construction of angle

Angles can be constructed using a compass and straight line, and a common task is to construct a specific angle, such as a 60 degree angle:

  1. Draw a base line of any length and mark a point A on it.
  2. Draw an arc on the line, placing the compass point at A.
  3. Without changing the width of the compass, place the compass at the intersection of the arc and the line, and draw a second arc that intersects the first arc.
  4. Draw a line from point A to the point of intersection of the arcs.
A

Example: You have constructed a 60 degree angle with angle BAC.

Construct a perpendicular line

A perpendicular line is constructed from a point on a line as follows:

  1. Place the tip of the compass at the given point on the line.
  2. Draw arcs on either side of the point that intersect the line without changing the width of the compass.
  3. From the intersection points on the line, draw arcs above and below the line to intersect each other.
  4. Draw a line through the origin and the intersection point of the new arcs.

Example: This method constructs a perpendicular line at a given point on the line.

Construct a parallel line

Constructing a parallel line through a given point using a line and compass:

  1. Draw an arc to intersect the line at the given point.
  2. Using the same compass width, draw another arc on the line where you want the parallel line.
  3. Measure the arc distance between intersections with a compass.
  4. Duplicate the same arc distance on the other arc.
  5. Draw a line through the new intersection and the given point.
point

Example: This shows the use of two arcs to construct a parallel line through a specified point.

Bisecting an angle

To bisect an angle means to divide it into two equal smaller angles:

  1. Draw an arc from the vertex of the angle that crosses both sides of the angle.
  2. From these intersection points, draw two arcs of equal length, which intersect each other.
  3. Draw a line from the vertex to the point of intersection of these new arcs to bisect the angle.

Example: This method is useful for constructing equal angles. Intersecting arcs guarantee accuracy.

Summary

Constructions in geometry provide a strong foundation for mathematical exercises. With a basic understanding of the construction of lines, angles, bisectors, and parallel and perpendicular lines, more complex shapes can also be constructed. This is not only an exercise in precision, but it also deepens the understanding of geometric facts and shapes.

Practice these steps with real tools like a compass and straightedge, and you'll find yourself mastering the art of geometric construction, and laying a solid foundation for further mathematical study.


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Constructions

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