Grade 7
  1. Number System  1.1. Integers  1.1.1. Properties of Integers  1.1.2. Operations on Integers  1.1.3. Additive and Multiplicative Inverses  1.1.4. Applications of Integers  1.2. Rational Numbers  1.2.1. Definition of Rational Numbers  1.2.2. Properties of Rational Numbers  1.2.3. Operations on Rational Numbers  1.2.4. Standard Form of Rational Numbers  1.3. Decimals  1.3.1. Operations on Decimals  1.3.2. Converting Between Fractions and Decimals  1.4. Powers and Exponents  1.4.1. Laws of Exponents  1.4.2. Simplifying Expressions with Exponents  1.4.3. Scientific Notation  1.5. Roots  1.5.1. Square Roots  1.5.2. Cube Roots
  2. Algebra  2.1. Expressions  2.1.1. Algebraic Expressions  2.1.2. Simplifying Expressions  2.1.3. Evaluating Expressions  2.2. Linear Equations  2.2.1. Solving Simple Equations  2.2.2. Applications of Linear Equations  2.3. Algebraic Identities  2.3.1. Expanding Using Identities  2.3.2. Simplifying Using Identities
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Simplifying Ratios  3.1.3. Equivalent Ratios  3.2. Proportion  3.2.1. Concept of Proportion  3.2.2. Direct Proportion  3.2.3. Inverse Proportion  3.3. Unitary Method  3.3.1. Solving Problems Using the Unitary Method
  4. Geometry  4.1. Lines and Angles  4.1.1. Types of Angles  4.1.2. Pairs of Angles  4.1.3. Properties of Parallel Lines and a Transversal  4.1.4. Angle Bisectors  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Angle Sum Property  4.2.3. Exterior Angle Property  4.2.4. Pythagoras Theorem  4.3. Congruence of Triangles  4.3.1. Criteria for Congruence  4.3.2. Applications of Congruence  4.4. Quadrilaterals  4.4.1. Types of Quadrilaterals  4.4.2. Properties of Parallelograms  4.4.3. Special Quadrilaterals  4.4.4. Properties of Rectangles and Rhombuses
  5. Mensuration  5.1. Perimeter and Area  5.1.1. Perimeter of Plane Figures  5.1.2. Area of Plane Figures  5.1.3. Area of a Trapezium  5.1.4. Area of a Parallelogram  5.1.5. Area of a Circle  5.1.6. Area of Composite Figures  5.2. Surface Area and Volume  5.2.1. Cubes and Cuboids  5.2.2. Surface Area of Cylinders  5.2.3. Volume of Prisms and Cylinders  5.2.4. Volume and Surface Area of Cones
  6. Data Handling  6.1. Data Collection  6.1.1. Organizing Data  6.1.2. Frequency Distribution  6.2. Graphical Representation of Data  6.2.1. Bar Graphs  6.2.2. Double Bar Graphs  6.2.3. Pie Charts  6.2.4. Histograms  6.2.5. Line Graphs  6.3. Probability  6.3.1. Simple Events  6.3.2. Experimental Probability  6.3.3. Theoretical Probability
  7. Practical Geometry  7.1. Construction of Triangles  7.1.1. Constructing Triangles Given Side Lengths  7.1.2. Constructing Triangles Given Side and Angle  7.1.3. Constructing Right-Angled Triangles  7.2. Construction of Quadrilaterals  7.2.1. Quadrilaterals Given Side Lengths and Angles  7.2.2. Constructing Parallelograms and Rectangles

Grade 7 → Practical Geometry ↓


Construction of Triangles


Practical geometry is an exciting aspect of mathematics where we learn to draw various shapes using some tools like ruler, compass, and protractor. Out of these shapes, triangle is a fundamental polygon that plays a vital role in both basic and advanced geometry. Triangulation involves drawing a triangle that obeys specific given conditions such as the length of sides or the measure of angles. Let's dive into the exciting world of drawing triangles!

Fundamentals of a triangle

A triangle has three sides, three angles and three vertices. The sum of the interior angles of a triangle is always 180 degrees. Based on the sides and angles, triangles are classified into different types such as:

  • Equilateral Triangle: All sides and angles are equal.
  • Isosceles triangle: Two sides and two angles are equal.
  • Scalene Triangle: All sides and angles are different.
  • Acute Triangle: All angles are less than 90 degrees.
  • Right Angle: An angle exactly 90 degrees.
  • Obtuse Triangle: One of its angles is more than 90 degrees.

Tools for drawing triangles

To make a triangle you will need the following basic tools:

  • Ruler: A straight tool for measuring and drawing straight lines.
  • Compass: An instrument for drawing circles or arcs and measuring the distance between two points.
  • Protractor: A semi-circular instrument for measuring and constructing angles.

Construction of a triangle given three sides (SSS)

Let's start drawing a triangle when the lengths of the three sides are given. Follow these steps:

  1. Draw a baseline on one side using your ruler.
  2. Using a compass, measure the other side. Place the compass needle at one end of the baseline and draw an arc.
  3. Place the same measurement on the compass for the third side, place the needle at the other end of the baseline, and draw another arc that crosses the first arc.
  4. The point where the arcs intersect is the third vertex of the triangle.
  5. Connect this vertex to the endpoints of your baseline to form a triangle.
Sides: a = 5 cm, b = 6 cm, c = 7 cm
phase: 
1. Draw line BC = 7 cm
2. Taking B as centre, draw an arc of radius 5 cm.
3. Taking C as centre, draw an arc of radius 6 cm.
4. The point of intersection is A
5. Connect A to B and A to C

Construction of a triangle given two sides and the angle between them (SAS)

To construct a triangle when you know two sides and the angle between them:

  1. Draw one of the given sides as the base line.
  2. Use the protractor to measure and draw a given angle from one end point of the baseline.
  3. Using a compass, measure the other side and draw an arc from the other end of the baseline that crosses the newly drawn angle line.
  4. The intersection point is the third vertex. Draw a triangle by connecting this point to the end points of the base lines.
Sides: a = 6 cm, b = 8 cm, ∠ = 60°
phase:
1. Draw line AB = 6 cm.
2. Measure and draw a 60° angle at A
3. Draw an arc of radius 8 cm from B which cuts the angle line
4. Name the point of intersection C 
5. Connect C to A and C to B

Construction of a triangle given two angles and the included side (ASA)

Construct a triangle when you are given two angles and the side between them:

  1. Draw the given side as the base line.
  2. Use the protractor to measure one of the given angles from one of the end points of the base.
  3. Do the same for the other angle from the other end point.
  4. The intersection point of the two lines is the third vertex. Connect this point to the ends of the base lines to complete the triangle.
Side: a = 5 cm, Angle: ∠A = 45°, ∠B = 60°
phase:
1. Draw line AB = 5 cm.
2. Draw a 45° line at A
3. Draw a 60° line at point B
4. The point of intersection is C
5. Connect C to A and C to B

Construction of a triangle given two angles and one non-corresponding side (AAS)

Constructing a triangle when given two angles and one non-corresponding side:

  1. Draw the given side as the base line.
  2. Measure and construct a given angle at one endpoint of the arm.
  3. Construct another angle with the other endpoint.
  4. Where the lines cross is the third vertex. Connect this point to the ends of the base lines to complete the triangle.
Side: b = 7 cm, Angles: ∠C = 50°, ∠A = 45°
phase:
1. Draw BC = 7 cm.
2. Draw a 50° line at point B
3. Draw a 45° line at C
4. The point of intersection is A
5. Connect A to B and A to C

Construction of a right-angled triangle (hypotenuse and one side)

To construct a right triangle with a given hypotenuse and one side:

  1. From the end point of the defined side, draw a right angle using the protractor.
  2. Use a compass to measure the length of the hypotenuse and draw an arc with one of the endpoints of the side as the center.
  3. Where the arc intersects the right angle line is your vertex point.
  4. Connect this point to complete the triangle.
Hypotenuse: 10 cm, Side: 6 cm
phase:
1. Draw line AB = 6 cm.
2. Measure 90° at B and draw the line upwards
3. Draw an arc of radius 10 cm from A on this line.
4. Mark the intersection with C
5. Join C to A and C to B

Conclusion

Constructing triangles revolves around some fundamental principles that depend on the sides of a triangle and its angles. Mastering construction techniques not only improves spatial understanding but also forms the basis for more advanced geometric explorations. Whether building an entire structure or solving everyday problems, knowing how to construct a triangle is an essential skill. Enjoy building!


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Construction of Triangles

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