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 → Triangles ↓


Pythagorean Theorem


The Pythagorean Theorem is a fundamental principle in mathematics, particularly in the study of geometry. This theorem is named after the ancient Greek mathematician Pythagoras. This theorem describes a special relationship between the sides of a right triangle. It states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. It is an important concept in mathematics that has various applications in daily life and other fields such as physics, engineering, and architecture.

Understanding right triangles

A right triangle is a type of triangle in which one angle measures exactly 90 degrees, called a right angle. It has three sides:

  • Hypotenuse: This is the longest side of the triangle, and it is always opposite the right angle.
  • Base: It is one of the two shorter sides that form a right angle.
  • Perpendicular: It is the other smaller side which along with the base helps to form a right angle.

Let's look at a visual representation of a right triangle:



    
    
    Base
    Straight
    Ear

Pythagorean theorem formula

The mathematical formulation of the Pythagorean theorem is expressed as follows:

 c² = a² + b²

Here:

  • c is the length of the hypotenuse.
  • a and b are the lengths of the other two sides (base and perpendicular).

How to use the Pythagorean theorem

The Pythagorean Theorem can be used to solve problems where you need to find the length of one side of a right triangle, while the lengths of the other two sides are known.

Example 1: Finding the hypotenuse

Suppose you have a right triangle whose base is 3 units long, and the perpendicular is 4 units long. We can find the length of the hypotenuse using the theorem.

 a = 3 b = 4 c² = a² + b² c² = 3² + 4² c² = 9 + 16 c² = 25 c = √25 c = 5

Therefore, the length of the hypotenuse is 5 units.

Example 2: Finding the missing side

What if we need to find the length of a missing side that is not the hypotenuse? Let's consider a triangle where the hypotenuse is 10 units, and the base is 6 units. We need to find the perpendicular.

 c = 10 a = 6 b = ? c² = a² + b² 10² = 6² + b² 100 = 36 + b² b² = 100 - 36 b² = 64 b = √64 b = 8

Therefore, the perpendicular side is 8 units long.

Interesting facts about the Pythagorean theorem

  • The Pythagorean Theorem only applies to right-angled triangles.
  • If you know the length of any two sides of a right triangle, you can always find the third side using the theorem.
  • This theorem can be used for real-world problem-solving, such as navigation, architecture, and physics.

Proofs of the Pythagorean theorem

There are several ways to show why the Pythagorean Theorem is true. Here, we present a simple algebraic proof:

Algebraic proof

 - Consider two squares. One is large with side (a+b) and the other smaller inside it of side c. - The large square can be divided into one smaller square of c² and four right-angled triangles each having sides a, b (base and perpendicular). Area of large square = (a+b)² = a² + b² + 2ab Area of 4 triangles = 4 × (1/2) × a × b = 2ab Thus, (a+b)² - 2ab = a² + b² = Area of smaller square, which is c² Therefore, c² = a² + b²

Practical applications of the Pythagorean theorem

  • Construction: Ensuring that structures are level and at the correct angles.
  • Navigation: Determining the minimum distance between points.
  • Art and design: Creating layouts that ensure symmetry and proportion.
  • Survey: Measuring land and determining property boundaries.

Extension of the Pythagorean theorem

Did you know that the Pythagorean Theorem has been generalized in various ways? For example, this theorem is a special case of the distance formula in coordinate geometry. In trigonometry, the theorem is the basis for defining the sine, cosine, and tangent functions for acute angles in a right triangle, relating angle measures to side lengths.

This theorem also forms the basis for the concept of Euclidean distance in coordinate systems beyond two dimensions, commonly known as multi-dimensional or n-dimensional Euclidean space.

Practice problems

Problem 1:

A right-angled triangle has one side 9 cm and the other side 12 cm. Find the length of the hypotenuse.

 a = 9 b = 12 c = ? c² = a² + b² c² = 9² + 12² c² = 81 + 144 c² = 225 c = √225 c = 15

The length of the hypotenuse is 15 cm.

Problem 2:

A ladder rests against a wall. The foot of the ladder is 7 m from the wall, and the ladder reaches a height of 24 m on the wall. Find the length of the ladder.

 base = 7 perpendicular = 24 hypotenuse = ? hypotenuse² = base² + perpendicular² hypotenuse² = 7² + 24² hypotenuse² = 49 + 576 hypotenuse² = 625 hypotenuse = √625 hypotenuse = 25

The length of the stairs is 25 meters.

Understanding and applying the Pythagorean Theorem is an essential skill when learning mathematics, providing insight into not only theoretical problems but also real-world applications. This theorem builds a bridge between algebraic concepts and geometric understanding, guiding students towards a deeper understanding of mathematical principles.


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