What is the significance of the 30-60-90 triangle?

The significance of the 30-60-90 triangle lies in its unique properties and the specific ratio of its sides, which is derived from the angles of the triangle. A 30-60-90 triangle has one angle measuring 30 degrees, another 60 degrees, and the last angle being 90 degrees, making it a right triangle.

In a 30-60-90 triangle, the lengths of the sides opposite these angles are in a fixed ratio. Specifically, the side opposite the 30-degree angle is the shortest and can be represented as 1 unit. The side opposite the 60-degree angle, which is longer than the 30-degree side, is √3 units long. Finally, the side opposite the 90-degree angle (the hypotenuse) is 2 units long. Therefore, the sides are in the ratio of 1 (shortest side) : √3 (longer side) : 2 (hypotenuse).

This consistent ratio allows for easy calculation and understanding of the triangle's dimensions in various applications, especially in trigonometry, geometry, and real-world problem-solving scenarios. In contrast to the other choices, which represent ratios from different types of triangles, the ratio of 1: