Right Triangle, Trigonometry, and the Pythagorean Theorem

In geometry, a right triangle is a triangle with one angle measuring exactly 90 degrees. Trigonometry, particularly in the context of right triangles, focuses on relationships between the lengths of its sides. There are three main types of sides in a right triangle: the adjacent side, the opposite side, and the hypotenuse. Each side is associated with an angle and is given a specific name based on that angle.

The hypotenuse is the longest side of a right triangle and is opposite the right angle. Since a right triangle has one right angle, there is only one place where the hypotenuse can be. Regardless of the direction you face, the hypotenuse will always be the longest side of the triangle. Its length is typically denoted as c or simply "side C".

The opposite side is not necessarily the side directly across from a particular angle; rather, it is the side on the opposite side of the designate angle. For example, if you consider angle A in the right triangle, the opposite side would be the side that is "across from" angle A, not necessarily directly across from it.

The adjacent side is the side that sits adjacent to the designate angle. Similar to the opposite side, the adjacent side isn't always immediately adjacent to the designated angle, but it shares a common vertex with the angle.

One fundamental concept related to these sides is the Pythagorean theorem. This theorem states that, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the adjacent and opposite sides. Mathematically, it is expressed as:

a^2 + b^2 = c^2

where a and b are the lengths of the adjacent and opposite sides, respectively, and c is the length of the hypotenuse.