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 ↓


Constructing Triangles


In geometry, drawing triangles is a basic skill. It involves constructing triangles using only a compass and a straight line. This skill helps in visualizing shapes and applying various geometric principles. This article will take you step-by-step through the process of constructing different types of triangles.

Introduction to triangles

A triangle is a three-sided polygon which is the simplest form of a closed figure. It has three vertices (corner points), three sides and three angles. The sum of the internal angles in a triangle is always 180 degrees. Based on the angles and the length of the sides, triangles can be classified into different types:

  • Equilateral triangle: All sides and angles are equal. Each angle is 60 degrees.
  • Isosceles triangle: Two sides are of equal length, and the angles opposite to these sides are also equal.
  • Scalene triangle: All sides and angles are different.
  • Right angle: One of its angles is 90 degrees.

Prerequisites for triangle construction

Before getting down to the construction methods, make sure you have the following tools:

  • Compass: For drawing arcs and circles.
  • Straightedge: A ruler without measurement markings, used for drawing straight lines.
  • Pencil: For sketching and drawing.
  • Pencil eraser: For corrections.

Understanding the basic construction steps

There are three primary methods for drawing triangles. Each method is based on different sets of three components: sides and angles.

Method 1: Side-side-side (SSS) construction

In this method, you will construct a triangle when the lengths of the three sides are known.

  1. Draw the base line: Use a straightedge to draw a line segment equal to the length of one side, say AB.
  2. Build the arch:
    • Open the compass to the length of the other side AC. Draw an arc using the compass with centre A.
    • Keep the third side BC equal to the length of the compass. With center B, draw another arc to intersect the first arc.
  3. Mark the intersection: Let C be the point of intersection of the arcs.
  4. Complete the triangle: Draw the line segments AC and BC using a straightedge.
    A─C
    
    │ B

Method 2: Side-angle-side (SAS) construction

This method covers cases where two sides and the angle between them are known.

  1. Draw the baseline: Start with a known side, say AB.
  2. Construct the given angle:
    • At point A, measure the given angle α using a protractor and mark it along the base line AB.
    • Draw a ray AC which makes an angle α with AB.
  3. Mark the other side:
    • Open the compass to the length of the other side AC. Place the compass at point A and cut the ray AC at point C.
  4. Draw the third side: Connect points B and C with a straight line.
    A
    
     C──B

Method 3: Angle-side-angle (ASA) construction

This technique is applicable where two angles and a side are known.

  1. Draw the known side: Start with the side, let's say AB.
  2. Construct an angle:
    • At point A, measure and draw angle α using the protractor.
  3. Construct the second angle:
    • Draw angle β using a protractor at point B.
  4. Extend both rays: Draw ray AC from A and ray BC from B. Intersect them at point C.
     C
        
     A──B

Special case construction

Let's take a look at some typical triangle formations that may arise:

Right-angled triangle construction

When drawing a right triangle, one angle is always 90 degrees:

  1. Create the hypotenuse: First create the hypotenuse, let's say AB.
  2. Construct a right angle:
    • Draw a line AC or BC by making an angle of 90 degrees at point A or B.
  3. Mark the other side:
    • Using a compass, mark point C where the second known side length intersects the ray.
  4. Connect the points: Connect the points AC and CB to complete the triangle.

Equilateral triangle construction

It is easy to draw an equilateral triangle because all sides are equal:

  1. Draw the side: Draw the side AB of the required length.
  2. Construct the circular arc:
    • Draw a circular arc with centre A and radius AB.
    • Draw another circle with centre B and the same radius.
  3. Mark the intersection: The point of intersection of the arcs is C.
  4. Connect the vertices: Draw lines AC and BC.
  C
     
A───B

Key concepts in triangle construction

Understanding the properties of triangles can enhance your construction skills:

1. Unique triangle structure

A unique triangle can be formed if certain conditions related to the sides and angles are met. The following conditions ensure a unique triangle:

  • Three Parties(S.S.S.)
  • Two sides and the angle between them (SAS)
  • Two angles and the side between them (ASA)

2. Triangle congruence

Triangle congruence plays an important role in the construction:

  • Two triangles are congruent if they have exactly the same size and shape.
  • Congruence terms include side-side-side (SSS), side-angle-side (SAS), angle-side-angle (ASA), etc.
By applying these congruence rules, reliable triangle construction becomes possible.

3. Applications in real-world problems

Triangular structures are used in many fields from architecture to engineering. Engineers use them to make plans and architects use them to design elements.

Conclusion

Drawing triangles is an essential skill in geometry that requires practice and precision. Mastering the various construction methods involves understanding fundamental properties such as congruence and uniqueness. This guide provides enough knowledge needed to accurately draw all types of triangles, enhancing your mathematical abilities.


Grade 9 → 11.3


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Constructing Bisectors
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Constructing Triangles

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