A triangle is a shape with three sides and three angles. A right triangle is a special kind of triangle that has one angle measuring exactly 90 degrees, like the corner of a square or a book. This 90-degree angle is called a right angle.

The two sides that form the right angle are called the legs. The third side, which is always the longest and is found opposite the right angle, is called the hypotenuse. Knowing these names is the first step to understanding the relationships within the triangle.

Over 2,000 years ago, Greek mathematicians discovered a special relationship between the sides of a right triangle. This relationship, known as the Pythagorean theorem, is a fundamental rule in geometry.

If we call the legs a and b and the hypotenuse c, we can write this relationship as a simple formula.

This formula allows you to find the length of any side of a right triangle if you know the lengths of the other two sides. Let's look at a classic example: a triangle with legs of length 3 and 4.

Using the theorem, we can find the length of the hypotenuse: $3^2 + 4^2 = c^2$ $9 + 16 = c^2$ $25 = c^2$ $\sqrt{25} = c$ $5 = c$

The hypotenuse is 5 units long. This 3-4-5 combination is a well-known example of a right triangle.

While the hypotenuse is always the hypotenuse, the legs have other names that depend on your point of view. Specifically, their names change depending on which of the two smaller angles (the acute angles) you are looking from.

Let's pick one of the acute angles and call it $\theta$ (theta).

• The opposite side is the leg directly across from the angle $\theta$.
• The adjacent side is the leg next to the angle $\theta$ that is not the hypotenuse.

These labels are crucial. If you switch to the other acute angle, the side that was opposite is now adjacent, and the side that was adjacent is now opposite. The hypotenuse, however, always stays the same.

Mastering these three concepts—the definition of a right triangle, the Pythagorean theorem, and how to identify the sides relative to an angle—provides the foundation for everything else in trigonometry.

Ready to test your knowledge? Let's see what you've learned.

With these basics down, you're ready to explore how angles and side lengths are connected.