A right triangle is a triangle with one interior angle measuring exactly 90 degrees. The side opposite the 90-degree angle is called the hypotenuse, while the other two sides are referred to as the legs of the triangle.

Pythagorean Theorem:

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs of the triangle).

In mathematical terms, if 'a' and 'b' represent the lengths of the legs of the right triangle and 'c' represents the length of the hypotenuse, the Pythagorean theorem can be expressed as:

c^2 = a^2 + b^2

This theorem provides a fundamental relationship between the three sides of a right triangle.

Solving a Right Triangle:

Given the lengths of any two sides of a right triangle, the Pythagorean theorem can be used to determine the length of the third side.

Example:

Suppose you have a right triangle with legs of length 3 units and 4 units. To find the length of the hypotenuse (c), you can apply the Pythagorean theorem:

c^2 = 3^2 + 4^2

c^2 = 9 + 16

c^2 = 25

c = √25

c = 5

Therefore, the hypotenuse of the right triangle is 5 units long.

Applications:

The Pythagorean theorem has numerous practical applications in various fields, including geometry, engineering, architecture, surveying, and navigation. It allows for the calculation of distances, angles, and other measurements related to right triangles, which are commonly encountered in real-world scenarios.