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 ↓


Constructing Triangles


In geometry, a triangle is one of the simplest shapes you can draw, yet it has endless possibilities and applications. Drawing a triangle is a fundamental skill in geometry that involves drawing an accurate triangle using given measurements. This guide will walk you through the process, rules, and methods for drawing a triangle.

Understanding triangle construction

Triangle construction involves drawing a triangle whose sides and angles satisfy specific conditions. To construct a triangle, you need some information about the triangle, which is usually enough to uniquely determine it.

Prerequisites

There are several conditions that can uniquely define a triangle, they are:

  • Three sides (SSS - Side-Side-Side)
  • Two sides and included angle (SAS - Side-Angle-Side)
  • Two angles and the included side (ASA - Angle-Side-Angle)
  • Two angles and a side (AAS - Angle-Angle-Side)

Impossible situations

Some conditions may appear to specify a triangle, but they are insufficient or vague:

  • Two sides and one non-inclusive angle (SSA - Side-Side-Angle) can form zero, one, or two triangles.
  • The three angles (AAA - angle-angle-angle) define the shape of the triangle, but not its scale.

Methods of constructing triangles

SSS (Side-Side-Side)

Drawing a triangle when three sides are known involves the following steps:

  • Step 1: Draw the longest side using the ruler.
  • Step 2: With the compass set to the length of the second side, draw an arc from one end of the first side.
  • Step 3: With the compass set to the length of the third side, draw another arc from the other end of the first side. The point where the arcs intersect is the third vertex of the triangle.

Example: Draw a triangle with sides 5 cm, 4 cm and 3 cm

Let us construct a triangle with sides 5 cm, 4 cm and 3 cm.

A B C

SAS (Side-Angle-Side)

This method is used when two sides and the included angle are known:

  • Step 1: Draw one side from the given sides.
  • Step 2: Use a protractor to measure the given angle from one endpoint of the side.
  • Step 3: From the same endpoint, draw the other arm at the angle measured. This gives a ray through which the arm will extend.
  • Step 4: Set the compass to the second given side length, then draw an arc to intersect the ray formed in Step 3. This intersection is the third vertex.

Example: Draw a triangle with sides 5 cm, 4 cm, and angle 60°

A B C

ASA (Angle-Side-Angle)

For this method, the two angles and the included side must be known:

  • Step 1: Draw the given side.
  • Step 2: Use the protractor to measure one of the given angles at one end of the side.
  • Step 3: Draw a ray at this angle from the endpoint.
  • Step 4: Measure the second angle at the other endpoint and draw the second ray.
  • Step 5: The point where the two rays intersect is the third vertex.

Example: Construct a triangle with angles of 30 °, 60 ° and side 5 cm

A B C

AAS (Angle-Angle-Side)

This method requires two angles and one unconnected side:

  • Step 1: Draw the known side.
  • Step 2: Measure an angle at one endpoint and draw a ray.
  • Step 3: Measure the second angle from the side and draw the second ray.
  • Step 4: The point of intersection of the rays is the third vertex of the triangle.

Example: Construct a triangle with angles of 45 °, 75 ° and side 6 cm

A B C

Common errors in triangle construction

  • Miscalculating angles: Even a slight mistake in measuring angles can lead to incorrect triangles. Always double-check your angles with a protractor.
  • Incorrect measurement of arms: Always use an accurate ruler and compass to ensure correct arm length.
  • Not using the right tools: Using makeshift tools can lead to mistakes. It is always best to use geometric tools designed for accuracy.

Exploring the properties of triangles

When drawing triangles, you can explore various properties and theorems related to them:

  • Triangle inequality theorem: The sum of the lengths of any two sides of a triangle is always greater than the length of the third side. This must be taken into account for the given dimensions to ensure the triangle truly exists.
  • Congruence and similarity: Constructing triangles helps in understanding when two triangles are congruent (exactly the same) or similar (same shape but different sizes).
  • Types of triangles: By constructing triangles, you can explore different types such as equilateral, isosceles, scalene, as well as right-angled, acute-angled, and obtuse-angled triangles.

Practice building triangles

The best way to master triangle construction is practice. Here are some exercises to develop your skills:

  1. Construct a triangle with sides 6 cm, 8 cm and 10 cm.
  2. Construct a triangle with base 7 cm having angles of 45° and 60° at each end point of the base.
  3. Construct a triangle with side length 5 cm, side length 5 cm and angle 90°. Identify the type of triangle formed.
  4. Construct a triangle with angles 35°, 55°, and the side between them is 4 cm. Verify the angle sum property of triangles.
  5. Construct an equilateral triangle whose each side measures 5 cm.

By doing these exercises, you will become proficient at using the techniques and principles of triangle construction. Understanding and mastering the art of triangle construction can be very beneficial in the broader study of geometry.

Conclusion

Drawing triangles is a core aspect of learning geometry, providing information about the fundamental properties and rules that define mathematical shapes. Through various construction methods like SSS, SAS, ASA and AAS, we can draw different types of triangles and understand their unique properties. As you practice these constructions, be aware of common pitfalls and apply the principles you learn to ensure accuracy.


Grade 8 → 3.2.4


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Constructing Triangles

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