The hypotenuse in Pythagorean Theorem is the longest side of a right-angled triangle, positioned opposite the right angle.

Pythagorean Theorem is a fundamental concept in geometry that specifically pertains to right-angled triangles. The theorem states that in such a triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This relationship is mathematically represented as

where $c$ denotes the length of the hypotenuse, and $a$ and $b$ represent the lengths of the other two sides.

To illustrate this, consider a right-angled triangle with side lengths of $3$ units and $4$ units. According to Pythagorean Theorem, we can determine the hypotenuse by calculating the square root of the sum of the squares of these two sides. Thus, we compute:

Consequently, the length of the hypotenuse $c$ is

Grasping the concept of the hypotenuse is essential, as it facilitates the resolution of various problems involving right-angled triangles, such as calculating distances, heights, and depths in diverse applications. Additionally, it serves as a foundational principle for more advanced topics in trigonometry and geometry. Remember, the hypotenuse is always the side opposite the right angle and is consistently the longest side in a right-angled triangle.