What is the relationship between the sides of a 30-60-90 triangle?

In a 30-60-90 triangle, the sides are proportional to specific ratios that correspond to the angles of the triangle. The side opposite the 30-degree angle is the shortest and is typically considered to be 1 unit. The side opposite the 60-degree angle is longer and is equal to ( \sqrt{3} ) units, while the hypotenuse, which is opposite the right angle, is twice the length of the shortest side, measuring 2 units.

Therefore, the sides of a 30-60-90 triangle are in the ratio of 1 (for the side opposite the 30-degree angle) to ( \sqrt{3} ) (for the side opposite the 60-degree angle) to 2 (for the hypotenuse). This results in the ratio of sides being 1:√3:2, establishing the unique relationship that characterizes this type of triangle.

This understanding is fundamental in solving problems involving right triangles, as it allows for the determination of side lengths based on the given angle measures without needing the actual dimensions.