All 30°-60°-90° Right Triangles are formed by taking half of a Equilateral Triange, as shown in the steps below.
Because the original triangle is Equilateral, that means all three sides are the same length. This is what variable "x" is trying to tell you. All three sides are the same length. Therefore, looking at diagram two, when you cut the triangle in half, new dimensions exist within the new triangle formed.
Since the original side length at the bottom of the Equilateral Triangle is cut in half, the length of that shorter side will always be half of the orignal lenth, x.
The angle at the top of the Equilateral Triangle is cut in half. Therefore the new degrees of the angle at the top of the triangle is half of 60°, or 30°.
Using the Pythagorean Theorem to solve for the last side length, and simplifying all square roots by hand, the third side ALWAYS has a length of the original side length, x, multiplied by the square root of 3. We will use this pattern to solve for the triangle without having to use Pythagorean Theorem every time.
All we need to solve a 30°-60°-90° Triangle is one of the side lengths. Then we can use the relationship between the three side lengths to solve for the other 2.
Above shows some examples of 30°-60°-90° triangles that have been solved for all three sides. Notice the pattern within each triangle's three side lengths. We can solve the trianble just by looking at it and using the patterns;no math involved.
