first of all pyhtagoras theorem formula is h²=p²+b² it can only be used on a right angle triangle the longest side of a triangle is called hypotenuse and shortest is called a base and third side is called a perpendicular so,the sum of the two sides ( perpendicular ²+base²)= hypotenuse²​

Pythagoras Theorem:

Pythagoras theorem, also known as Pythagorean theorem, is a fundamental concept in mathematics that relates the sides of a right-angled triangle. It establishes a relationship between the lengths of the three sides of any right triangle, and it is named after the ancient Greek mathematician Pythagoras.

The Theorem:

The Pythagorean theorem states that in any 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. Mathematically, it can be expressed as:

c² = a² + b²

Where:
- c represents the length of the hypotenuse.
- a and b represent the lengths of the other two sides.

Explanation:

To understand the Pythagorean theorem, let's consider a right-angled triangle with sides a and b and hypotenuse c. By squaring the lengths of the sides, we get a² and b², respectively. According to the theorem, the sum of these squares should be equal to the square of the hypotenuse, which is c².

Example:

Let's take an example where a = 3 and b = 4. Using the Pythagorean theorem, we can find the length of the hypotenuse. Plugging the values into the equation:

c² = 3² + 4²
c² = 9 + 16
c² = 25

Therefore, c = 5. So, the length of the hypotenuse is 5 units.

Applications:

The Pythagorean theorem has numerous applications in various fields, including:

1. Construction: Architects and engineers use the theorem to ensure the accuracy of right angles in building structures.
2. Navigation: It is used in navigation systems to calculate distances and angles.
3. Physics: The theorem is used in physics, particularly in vector analysis and calculating forces.
4. Trigonometry: The Pythagorean theorem is a fundamental concept in trigonometry, forming the basis for many trigonometric identities and equations.

Conclusion:

The Pythagorean theorem is a vital mathematical principle that provides a relationship between the sides of a right-angled triangle. It has diverse applications in different fields, making it an indispensable tool in various disciplines. Understanding and utilizing the theorem allows for precise calculations and problem-solving in numerous real-life situations.