Prove. Pythagoras theorem?

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What is Pythagoras theorem?

Pythagoras theorem is a fundamental result in geometry 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 lengths of the other two sides.

Pythagoras theorem formula:

The formula for Pythagoras theorem can be written as:

c² = a² + b²

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

Proof of Pythagoras theorem:

There are several methods to prove Pythagoras theorem, but one of the most commonly used and simple proofs is the geometric proof.

Geometric Proof:

1. Start with a right-angled triangle with sides a, b, and hypotenuse c.
2. Construct squares on each side of the triangle. The area of the square on side a is a², the area of the square on side b is b², and the area of the square on side c is c².
3. Now, divide the square on side c into two rectangles, one with dimensions a x a and the other with dimensions b x b.
4. Arrange these two rectangles inside the square on side c, so that they completely fill the square without any overlap.
5. It can be observed that the resulting figure is a square with side length (a + b).
6. By comparing the areas, we can conclude that the area of the square on side c (c²) is equal to the sum of the areas of the squares on sides a (a²) and b (b²).
7. Therefore, the equation c² = a² + b² holds true for any right-angled triangle.

Conclusion:

Through the geometric proof, we have successfully demonstrated that the Pythagoras theorem is valid for all right-angled triangles. This theorem has immense applications in various fields of mathematics and science, making it an essential concept to understand.

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