In mathematics, the Pythagorean theorem, also known as Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":[1]

Pythagoras Theorem:

The Pythagoras Theorem is an essential concept in mathematics that relates to the sides of a right-angled triangle. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

The theorem can be mathematically represented as:

c^2 = a^2 + b^2

Where:
- "c" represents the length of the hypotenuse
- "a" and "b" represent the lengths of the other two sides (known as the legs) of the right triangle.

Explanation:

The Pythagoras Theorem can be explained in detail through the following points:

1. Right-angled Triangle: The Pythagoras Theorem applies to right-angled triangles, which have one angle measuring 90 degrees. This angle is known as the right angle.

2. Hypotenuse: The hypotenuse is the longest side of a right-angled triangle and is always opposite the right angle.

3. Legs: The other two sides of the right-angled triangle are called legs. These sides form the right angle.

4. Relationship: According to the theorem, the square of the hypotenuse's length is equal to the sum of the squares of the lengths of the other two sides.

5. Mathematical Representation: The theorem can be expressed as c^2 = a^2 + b^2, where "c" is the length of the hypotenuse, and "a" and "b" are the lengths of the legs.

6. Example: Consider a right-angled triangle with side lengths of 3 units and 4 units. Using the Pythagoras Theorem, we can find the length of the hypotenuse. Substituting the values in the equation, we have c^2 = 3^2 + 4^2, which simplifies to c^2 = 9 + 16. Ultimately, we find that c^2 = 25, and taking the square root on both sides, c = 5 units.

7. Applications: The Pythagoras Theorem is widely used in various fields such as engineering, architecture, navigation, physics, and trigonometry. It helps in calculating distances, determining unknown lengths in triangles, and solving real-world problems involving right-angled triangles.

Conclusion:

The Pythagoras Theorem is a fundamental concept in mathematics that provides a relationship between the sides of a right-angled triangle. It allows us to find the length of the hypotenuse or the unknown sides of a triangle, making it a valuable tool in various practical applications.