The Pythagorean Theorem (a² + b² = c²)

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The Pythagorean Theorem is a fundamental concept in geometry and trigonometry, expressed as \(a^2 + b^2 = c^2\), where \(c\) is the length of the hypotenuse of a right-angled triangle, and \(a\) and \(b\) are the lengths of the other two sides. This theorem provides a universal relationship between the sides of any right-angled triangle, allowing for the calculation of unknown sides when the other two are known. It is a crucial tool in various mathematical and real-life applications, including architecture, engineering, and navigation. The theorem's significance extends beyond its mathematical formula. It is reversible, meaning if \(a^2 + b^2 = c^2\), then the triangle is right-angled. This property helps classify triangles into acute or obtuse based on the relationship between their sides. The theorem has been proven in numerous ways throughout history, with Euclid's proof being one of the most notable. Its practical applications make it a cornerstone of geometric calculations and problem-solving.