Hypotenuse

Hypotenuse

Triangle Sides ABC

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Hypotenuse is the longest side of a right-angled triangle, opposite the right angle. It is a fundamental concept in geometry, particularly in Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This relationship is expressed as \(c^2 = a^2 + b^2\), where \(c\) represents the length of the hypotenuse, and \(a\) and \(b\) represent the lengths of the triangle's other two sides.

History

The concept of the hypotenuse and the Pythagorean theorem can be traced back to ancient civilizations, including the Babylonians, Egyptians, and Indians, but it is most commonly associated with the ancient Greek mathematician Pythagoras. Although there is no concrete evidence that Pythagoras himself developed the theorem, it was proven and formalized by his followers and later mathematicians.

Applications

The hypotenuse is crucial in various fields such as engineering, architecture, and physics, where it is used to calculate distances, forces, and other properties in scenarios involving right-angled triangles. In trigonometry, the lengths of the hypotenuse and other sides of right-angled triangles are used to define the sine, cosine, and tangent functions, which are essential for solving problems involving angles and distances.

Calculating the Hypotenuse

To calculate the length of the hypotenuse given the lengths of the other two sides, one can use the Pythagorean theorem. For example, if the lengths of the other two sides are 3 units and 4 units, the length of the hypotenuse can be calculated as follows:

\[ c = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

Thus, the hypotenuse would be 5 units long.

See Also

• Triangle
• Right-angled triangle
• Pythagorean theorem
• Trigonometry

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