The hypotenuse, denoted as “c,” is a key concept in geometry and trigonometry. It is an important component of the 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.

The hypotenuse is the longest side in a right-angled triangle and is always opposite the right angle. This means that it is the side that is directly across from the 90-degree angle. In a right-angled triangle with sides of length “a” and “b,” the hypotenuse can be found using the formula:

c = √(a^2 + b^2)

This formula is derived from the Pythagorean theorem, which is one of the fundamental principles of geometry. It is named after the ancient Greek mathematician Pythagoras, who is credited with discovering this relationship between the sides of a right-angled triangle.

The concept of the hypotenuse is not limited to right-angled triangles, as it can also be applied in the context of trigonometry. In a right-angled triangle, the hypotenuse is related to the other two sides by the trigonometric functions sine, cosine, and tangent. These functions provide a way to calculate the relationship between the angles and sides of a right-angled triangle, with the hypotenuse serving as a key reference point.

The hypotenuse is also important in various real-world applications, such as architecture, engineering, and physics. For example, when constructing buildings, engineers use the concept of the hypotenuse to ensure the structural stability and integrity of the edifice. In physics, the hypotenuse is utilized in calculations related to forces, vectors, and motion.

In conclusion, the hypotenuse is a fundamental element in geometry and trigonometry, playing a crucial role in the study of right-angled triangles and their applications in various fields. Understanding the concept of the hypotenuse and its relationship to the other sides of a right-angled triangle is essential for anyone studying mathematics, science, or engineering.