What are the ratios of a 30-60-90 special right triangle?

In a 30-60-90 special right triangle, the ratios of the lengths of the sides are distinctly defined based on the angles. The side opposite the 30-degree angle is the shortest and can be represented as (x). The side opposite the 60-degree angle, which is longer than the 30-degree side, is (x\sqrt{3}). Finally, the hypotenuse, which is opposite the right angle, is the longest side and is represented as (2x).

This pattern emerges from the properties of a 30-60-90 triangle, where the lengths adhere to the formula derived from the geometric relationships and the Pythagorean theorem. By knowing that the shortest side is (x), you can easily find the other two sides:

• The hypotenuse is always double the shortest side, leading to the expression (2x).

• The longer leg, which corresponds to the 60-degree angle, has a length that is (\sqrt{3}) times the length of the shortest side, resulting in (x\sqrt{3}).

Thus, the ratios (x), (x\sqrt{3}), and (2x) represent the sides of