Final answer:
The Pythagorean Theorem, developed by Pythagoras, states that in a right-angled triangle, the sum of the areas of the squares on the two shorter sides (a and b) is equal to the area of the square on the hypotenuse (c), expressed as a² + b² = c². This geometric principle is a foundational concept in mathematics and plays a critical role in various practical applications such as navigation and construction.
Step-by-step explanation:
Pythagorean Theorem According to Pythagoras
Pythagoras's version of the Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This relationship is represented as the equation: a² + b² = c². To find the length of the hypotenuse, the equation can be rewritten as: c = √a² + b². This profound insight not only underpins geometry but also demonstrates how the natural world, like distances and shapes, can be described mathematically.
The Pythagoreans went beyond mathematics to explore the nature of the universe, believing in a harmonious structure governed by numeric relationships. They applied their understanding to various fields, including music, astronomy, and the philosophy of the soul. Here is the visual representation of the theorem, where the right triangle has legs 'a' and 'b', and the hypotenuse is labeled 'c':
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c|
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a b
The areas of squares built on the legs (a and b) add up to the area of the square built on the hypotenuse (c), meaning that the total area can be expressed using the formula: Area of square a + Area of square b = Area of square c. This principle is incredibly useful, for example, in navigation or construction, where the shortest path or structural dimensions must be calculated precisely.