In a 30-60-90 triangle, what is the ratio of the sides?

In a 30-60-90 triangle, the sides have a specific ratio that can be derived from its properties. The main characteristics of a 30-60-90 triangle are that the angles measure 30 degrees, 60 degrees, and 90 degrees. The side opposite the 30-degree angle is the shortest side, often referred to as 'x'. The side opposite the 60-degree angle is longer and can be expressed as 'x√3', while the side opposite the 90-degree angle (the hypotenuse) measures '2x'.

To establish the ratio of the sides, we start with the shortest side (opposite the 30-degree angle), followed by the longer leg (opposite the 60-degree angle), and finally the hypotenuse. This gives us the ratio:

• Shortest side (opposite 30 degrees): x

• Longer side (opposite 60 degrees): x√3

• Hypotenuse (opposite 90 degrees): 2x

When you express the ratio of the sides in simplest form, it becomes 1 (for the shortest side) : √3 (for the longer side) : 2 (for the hypotenuse). This leads to