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The Pythagorean theorem, attributed to Pythagoras, is a fundamental principle in Euclidean geometry. It states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is crucial for solving geometric problems, understanding trigonometric ratios, and identifying Pythagorean triples, which are sets of three integers that form the sides of a right-angled triangle. Its applications span across construction, navigation, physics, and various scientific and engineering disciplines.

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## Definition and History of the Pythagorean Theorem

### Origin and Attribution

The Pythagorean theorem is a fundamental concept in geometry, attributed to the ancient Greek philosopher and mathematician Pythagoras

### Algebraic Expression

Variables and Equations

The Pythagorean theorem is expressed algebraically as c² = a² + b², where 'c' represents the length of the hypotenuse and 'a' and 'b' represent the lengths of the other two sides

### Visual Representation

The Pythagorean theorem can be visualized by constructing squares on each side of a right-angled triangle, demonstrating the relationship between the squares' areas

## Applications of the Pythagorean Theorem

### Real-World Problem Solving

The Pythagorean theorem is a practical tool for solving problems involving right-angled triangles, such as in construction, navigation, and physics

### Trigonometry

Trigonometric Ratios

The Pythagorean theorem is foundational to the study of trigonometry, which involves the relationships between the angles and sides of triangles

### Pythagorean Triples

Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem, commonly used in practical applications involving right-angled triangles

## Significance and Legacy of the Pythagorean Theorem

### Fundamental Principle of Geometry

The Pythagorean theorem is a fundamental principle of geometry that establishes a relationship between the sides of a right-angled triangle

### Mathematical Education

The Pythagorean theorem is a vital component of mathematical education and continues to be widely used in various fields, emphasizing its importance and utility

### Enduring Formula

The enduring formula of the Pythagorean theorem, c² = a² + b², is a testament to the lasting impact of Pythagoras's mathematical legacy

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