Understanding 30-60-90 triangles

Hey, can someone help me understand what a 30-60-90 triangle is and how to use it in math problems? Thanks in advance!

Sure, I'd be happy to help you understand 30-60-90 triangles! A 30-60-90 triangle is a special right-angled triangle where the angles measure 30 degrees, 60 degrees, and 90 degrees. Because the angles are fixed, the side lengths have a specific relationship, which makes it easy to solve problems involving this type of triangle.

The side lengths follow these ratios:

- The side opposite the 30-degree angle is the shortest side. Let's call this side "a."

- The side opposite the 60-degree angle is sqrt(3) times the shortest side. So, this side will be "a sqrt(3)."

- The side opposite the 90-degree angle, the hypotenuse, is twice the shortest side. So, the hypotenuse will be "2a."

Knowing these ratios can help you quickly determine the side lengths if you're given one side. Here's an example problem:

Suppose we have a 30-60-90 triangle with a hypotenuse of length 10. What are the lengths of the other two sides?

1. We know the hypotenuse is twice the shortest side. So, a = 10/2 = 5.

2. The side opposite the 30-degree angle is the shortest side, which we just found to be "a=5."

3. The side opposite the 60-degree angle is a sqrt(3) = 5 sqrt(3).

So, the side lengths are 5, 5√3, and 10. I hope this helps you understand 30-60-90 triangles and how to use them in math problems!