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 → Geometry ↓


Triangles


Triangles are one of the simplest and most interesting shapes in geometry. They are polygons with three edges and three vertices. The word "triangle" comes from the Latin words "tri" meaning three and "angulus" meaning angle. Let's learn all about triangles.

What is a triangle?

A triangle is a closed figure with three straight sides and three angles. The sum of these angles is always 180 degrees. You can think of a triangle as a slice of pizza, a musical instrument like a triangle, or a simple roof structure. Mathematically, a triangle can be defined as:

A triangle is a three-sided polygon with three vertices, three interior angles, and the sum of the internal angles is 180 degrees.

Types of triangles

Triangles can be classified into different categories based on their sides and angles.

Based on the sides

There are three types of triangles depending on the length of the sides:

  • Equilateral triangle: The three sides are equal in length and the three angles are equal, each measuring 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.

Based on angles

Triangles can also be classified according to their angles:

  • Acute triangle: All three angles are less than 90 degrees.
  • Right-angled triangle: One of its angles is equal to 90 degrees.
  • Obtuse triangle: One of its angles is more than 90 degrees.

Visualization of triangles

Let us look at the different types of triangles through simple diagrams:

Equilateral triangle

Isosceles triangle

Scalene triangle

Right triangle

Properties of triangles

Triangles have several important properties:

  • The sum of all the interior angles of any triangle is 180 degrees.
  • Equilateral triangles have equal sides and angles.
  • An isosceles triangle has two equal sides and the angles opposite to these sides are equal.
  • In a right-angled triangle, the sides obey the Pythagorean theorem.
  • The area of a triangle can be calculated using the following formula:
Area = (base * height) / 2

For example, the area of a triangle with a 10 unit base and 5 unit height would be:

Area = (10 * 5) / 2 = 25 square units

Applications of triangles

Triangles are not just theoretical shapes; they have practical applications in a variety of fields:

  • Architecture: Triangles are used in the design of bridges and buildings because they distribute weight evenly.
  • Engineering: Triangles provide structural strength in construction, making structures and foundations more stable.
  • Art and design: Artists and designers use triangles to create dynamic compositions and patterns.
  • Navigational instruments: Right triangles are used in navigation and to calculate distances in maps.

Explore more with triangles

Let's take a deeper look at some interesting mathematical concepts related to triangles:

Pythagorean theorem

The Pythagorean theorem applies to right-angled triangles. It states that:

a² + b² = c²

where a and b are the lengths of the two shorter sides, and c is the hypotenuse, the side opposite the right angle.

For example, in a right triangle with sides 3 units and 4 units, the hypotenuse can be calculated as:

3² + 4² = c² 9 + 16 = c² 25 = c² c = √25 c = 5 units

Triangle inequality theorem

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In mathematical terms:

a + b > ca + c > bb + c > a

For example, with 5, 7, and 10 unit sides, you could check:

5 + 7 > 10 5 + 10 > 7 7 + 10 > 5

Since all the inequalities are true, these sides can form a triangle.

Fun facts and additional information

  • The smallest angle of an equilateral triangle is always smaller than the smallest angle of any non-equilateral triangle.
  • No matter how much the triangle is folded or turned, the total angle always remains 180 degrees.
  • You can find triangles in nature, such as in spider webs or leaves with triangular patterns.

Conclusion

Triangles are fundamental shapes that form the basis of many structures and mathematical concepts. By understanding their properties, types, and applications, we can appreciate both their simplicity and complexity. Triangles are everywhere around us, supporting physical structures and connecting mathematical ideas.


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Triangles

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