Special right triangles include two specific types: the 45-45-90 triangle and the 30-60-90 triangle. Each of these triangles has a unique set of formulas for determining the lengths of their sides.
45-45-90 Triangle
In a 45-45-90 triangle, both angles are 45 degrees and the sides opposite these angles are equal in length. The formulas for finding the lengths of the sides are as follows:
- If one leg (the side opposite the 45-degree angles) is x, then the length of the hypotenuse (the side opposite the 90-degree angle) is x√2.
30-60-90 Triangle
In a 30-60-90 triangle, the angles are 30 degrees, 60 degrees, and 90 degrees. The side lengths are in a consistent ratio. The formulas for these triangles are:
- If the shortest side (the side opposite the 30-degree angle) is x, then the length of the side opposite the 60-degree angle is x√3, and the length of the hypotenuse is 2x.
These formulas are incredibly useful for quickly determining side lengths in right triangles during problem-solving in geometry. Remembering the properties of these special triangles can help simplify many mathematical calculations and assist in visualizing geometric concepts.