In a 30-60-90 triangle, what is the relationship between the lengths of the sides?

In a 30-60-90 triangle, there is a specific and consistent relationship between the lengths of the sides based on the angles present. The triangle's unique properties help to create ratio relationships that can be easily discerned.

The side opposite the 30-degree angle is indeed half the length of the hypotenuse. This means that if you know the length of the hypotenuse, you can simply divide it by two to find the length of the side opposite the 30-degree angle. For instance, if the hypotenuse measures 10 units, the side opposite the 30-degree angle would be 5 units.

The side opposite the 60-degree angle will be the longer leg of the triangle, and its length can be determined using the ratio: it is equal to the length of the side opposite the 30-degree angle multiplied by the square root of 3. This establishes a clear hierarchy where the hypotenuse is the longest side, the side opposite the 60-degree angle is longer than the side opposite the 30-degree angle but shorter than the hypotenuse, and the side opposite the 30-degree angle is the shortest.

This inferential structure illustrates why the option stating that the side opposite the 30