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 ↓


Basic Constructions


In mathematics, constructing means drawing shapes, angles or lines precisely. These constructions are often done with the help of tools like a compass, straightedge (ruler) or protractor. The beauty of constructing is in the precision and pure logic behind creating shapes and angles without any numerical measurements. This should give you a deeper understanding of geometry. Let's take a look at some basic constructions.

1. Construction of a line segment

A line segment is a portion of a line that is bounded by two distinct endpoints. Creating a line segment of a fixed length is a basic task. You can do it like this:

  1. Draw a straight line using a ruler.
  2. Mark a point A on the line. This will be one of the endpoints of the line segment.
  3. Place the compass pointer at A, and open it to the desired length of the segment.
  4. Keeping the compass open at the same width, place the pointer on A and draw an arc across the line.
  5. Mark the point where the arc and line intersect, B. AB is the line segment you created.
A B

2. Constructing a perpendicular bisector of a line segment

The perpendicular bisector of a line segment is the line that divides the segment into two equal parts at a 90 degree angle. Here is how to construct it:

  1. Given a line segment AB, place the compass pointer at one of its ends, say A, and adjust its width so that it covers a little more than half of the line segment.
  2. Draw arcs above and below the line.
  3. Without changing the width of the compass, repeat the same from the other endpoint B.
  4. Mark the two points P and Q where the arcs intersect.
  5. Draw a line through P and Q. This is the perpendicular bisector of line AB.
A B P Q

3. Construction of angle bisector

An angle bisector is a line that divides an angle into two equal smaller angles. Constructing it involves the following steps:

  1. Place the compass needle at point Y for angle ∠XYZ.
  2. Draw an arc on both sides of the angle, creating intersection point A on side XY and intersection point B on side YZ.
  3. Now, place the compass needle at A and draw an arc inside the angle.
  4. Without changing the width of the compass, repeat the same arc from point B.
  5. Mark the point of intersection of the two arcs as C.
  6. Draw a line from Y to C. This line is the bisector of ∠XYZ.
X Z Y A B C

4. Construction of perpendicular from a point to a line

To draw a perpendicular from a point to a line, follow these steps:

  1. Let the line be l and the point be P.
  2. Place the compass needle at P and draw an arc that cuts the line at two points A and B.
  3. With the compass set to the widest position, draw arcs above and below the line, starting at A and B.
  4. Mark the points of intersection of these arcs with C and D.
  5. Draw a line through P and CD. It is perpendicular to the line l from point P.
P A B C D

5. Construction of an equilateral triangle

All sides of an equilateral triangle are of equal length. This simple construction is as follows:

  1. Draw a line segment AB which will be one side of the triangle.
  2. Keeping the width of the compass equal to AB, place the pointer at A and draw an arc.
  3. Without changing the width, place the pointer on B and draw a second arc intersecting the first arc.
  4. Label the point of intersection C.
  5. Draw lines AC and BC to construct triangle ABC.
A B C

These are just basic constructions, but mastering them will give you a way to create more complex geometric drawings and better understand the properties of geometric shapes. Enjoy practicing these constructions and see the precision and satisfaction that comes from progressing through these mathematical operations. Geometry taught through constructions provides a deeper and more intuitive understanding of mathematical principles.

Understanding these fundamental methods will improve your ability to see shapes and understand geometry. In practice, the compass is a valuable tool for accurate construction; learning to use it well is important for anyone wishing to better understand the art of construction in mathematics. Learning through construction not only applies to mathematics, but develops a disciplined way of looking at problem-solving in everyday life.

Summary

Basic geometric constructions allow us to create geometric shapes without the need for measurements. We have explored several constructions:

  • Constructing a line segment
  • Perpendicular bisector
  • Angle bisector
  • Perpendicular from a point to a line
  • Equilateral triangle

Mastering these constructions lays the foundation for understanding geometric properties and relationships, rather than relying solely on measuring tools. These basics lend themselves to a deeper understanding of more advanced geometric concepts.

Practicing these skills requires patience and accuracy, it enhances your understanding of mathematical concepts and develops your logical thinking skills. Remember, the key to success in geometric constructions is practice, accuracy and patience.


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Basic Constructions

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