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 ↓


Pythagoras Theorem


The Pythagorean Theorem is one of the most important rules of geometry, which allows us to work with right-angled triangles. It is named after the ancient Greek mathematician Pythagoras. Although the theorem was known to people long before Pythagoras, he was the first to prove it rigorously mathematically.

Understanding right triangles

Before we dive into the Pythagorean Theorem, let's understand what a right triangle is. A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. The side opposite this angle is called the hypotenuse, and it is always the longest side of a right triangle. The other two sides are known as the "legs" of the triangle.

            A right triangle:

                  C
                ,
               ,
               ,
             |A |/|B
               ,

Pythagorean Theorem

The Pythagorean theorem provides a relationship between the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The theorem can be expressed as:

            c² = a² + b²

This equation helps us determine the lengths of the sides, if we know at least two of them.

Visual example

Consider a right triangle where side a = 3 units, side b = 4 units, and we want to find the length of the hypotenuse c.

              Respectively:
                  a² = 3² = 9
                  b² = 4² = 16

                  So, c² = a² + b²
                      = 9 + 16
                      = 25

                  Therefore, c = √25 = 5
C (5) One (3) B (4)

Working through examples

Example 1

Suppose you have another right triangle with one side a 5 units long and the hypotenuse c 13 units long. We need to find the length of the other side b.

              Respectively:
                  c² = 13² = 169
                  a² = 5² = 25

                  So, c² = a² + b²
                      169 = 25 + b²

                  Therefore, b² = 169 – 25
                             = 144

                  So, b = √144 = 12

Example 2

Suppose you know both legs of a right-angled triangle: a = 8 units and b = 6 units, and you want to determine the hypotenuse c.

              Respectively:
                  a² = 8² = 64
                  b² = 6² = 36

                  So, c² = a² + b²
                      = 64 + 36
                      = 100

                  Therefore, c = √100 = 10

When to use the Pythagorean Theorem?

The Pythagorean Theorem is incredibly useful not only in pure mathematics, but in many practical situations. It is commonly used to solve problems involving distance and measurement in construction, navigation, and physics.

Here are some scenarios where the theorem can be applied:

  • Calculating the shortest path between two points (e.g., the distance along a straight line).
  • To find the distance covered by a ladder placed at an angle along the wall.
  • Determining the size of a TV screen given its height and width.

Proof of the Pythagorean theorem

To prove the Pythagorean theorem, it is necessary to understand why this relation is true for right-angled triangles. There are several proofs, but the most popular is the geometric proof, which uses rearrangements and algebra:

Consider the areas of the squares built on each side of a right-angled triangle. The total area of the two smaller squares (built on sides a and b) is equal to the area of the largest square built on the hypotenuse c. This shows why the sum of the squares of the legs is equal to the square of the hypotenuse.

Concluding remarks on the Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that opens up a world of mathematical exploration. This theorem not only serves as a bridge to higher-level mathematics, but is also the basis for many technologies and sciences. By mastering this concept, you equip yourself with the tools to analyze and solve a wide variety of geometric problems.

Whether you're bridging gaps in construction, traversing space, or simply studying geometry, the Pythagorean Theorem gives you the foundation you need to understand measurements and distances in a rectangular world.


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