**Pythagoras Theorem**

**Introduction:**
Pythagoras Theorem is a fundamental concept in mathematics that relates to the sides of a right-angled triangle. It is named after the ancient Greek mathematician Pythagoras, who is credited with its discovery. The theorem 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 lengths of the other two sides.

**The Theorem:**
The Pythagorean Theorem can be stated mathematically as:
c² = a² + b²
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides of the right-angled triangle.

**Proof of the Theorem:**
There are various mathematical proofs to establish the validity of the Pythagorean Theorem. One of the most common proofs is the geometric proof, which involves using squares to demonstrate the relationship between the sides of a right-angled triangle.

**Geometric Proof:**
1. Start with a right-angled triangle ABC, where angle C is the right angle.
2. Construct squares on each side of the triangle, forming squares ABDE, BCFG, and ACHI.
3. The area of square ABDE is equal to the sum of the areas of squares BCFG and ACHI.
4. Using the formulas for the areas of squares, we can derive the equation:
(a + b)² = a² + b² + 2ab
5. Simplifying this equation gives:
a² + b² = (a + b)² - 2ab
6. Since the areas of squares ABDE, BCFG, and ACHI are equal, we can equate their respective side lengths:
c² = a² + b²
7. Thus, the Pythagorean Theorem is proven.

**Application of the Theorem:**
The Pythagorean Theorem has numerous applications in mathematics, physics, engineering, and everyday life. Some common applications include:
- Calculating the distance between two points in a coordinate plane.
- Determining the length of a diagonal or a side of a rectangle.
- Solving problems involving right-angled triangles, such as finding missing sides or angles.
- Applying trigonometry to solve real-world problems, such as measuring distances or heights using angles and distances.

In conclusion, the Pythagorean Theorem is a fundamental concept in mathematics that relates to the sides of a right-angled triangle. It has a wide range of applications and is proven through various mathematical proofs, including the geometric proof. Understanding and applying this theorem is crucial in many fields of study and everyday situations.