Triangles in Geometry 

A 30-60-90 triangle is a special right triangle where the sides are in the ratio of 1:√3:2. The angle between the longer side and the hypotenuse is always 60°, and the angle between the shorter side and the hypotenuse is always 30°. Let's explore a few properties of 30-60-90 triangles.

Properties of 30-60-90 Triangles 

There are a few key properties that all 30-60-90 triangles share. First, as we mentioned before, the sides of these triangles will always be in the ratio of 1:√3:2. This means that if you know the length of one side, you can easily calculate the lengths of the other two sides. For example, if the length of Side A is 3, then the length of Side B will be 3√3 (or approximately 5.196), and the length of Side C will be 6. 

Another important property to note is that all 30-60-90 triangles will have one acute angle and two obtuse angles. In other words, one angle will measure less than 90°, while the other two angles will measure more than 90°. You can also use this property to determine whether a triangle is a 30-60-90 triangle—if a triangle has one acute angle and two obtuse angles, it's a good bet that it's a 30-60-90 triangle! 

Conclusion

The 30-60-90 triangle is a special right triangle with some unique properties. If you ever come across a triangle with one acute angle and two obtuse angles, there's a good chance it's a 30-60-90 triangle. These triangles also have sides that are in the ratio of 1:√3:2, which means that if you know the length of one side, you can easily calculate the lengths of the other two sides. Knowing how to identify and work with 30-60-90 triangles can definitely come in handy in geometry class!

FAQ

What is a 30-60-90 triangle definition?

A 30-60-90 triangle is a special right triangle where the sides are in the ratio of 1:√3:2. The angle between the longer side and the hypotenuse is always 60°, and the angle between the shorter side and the hypotenuse is always 30°.

What is the formula for 30-60-90 triangle?

The formula for a 30-60-90 triangle is 1:√3:2. This means that if you know the length of one side, you can easily calculate the lengths of the other two sides.

How do you prove a 30-60-90 triangle?

To prove that a triangle is a 30-60-90 triangle, you need to show that the sides are in the ratio of 1:√3:2 and that one angle is 60°, one angle is 30°, and one angle is 90°. You can also use the fact that all 30-60-90 triangles will have one acute angle and two obtuse angles.

What are the lengths of a 30-60-90 triangle?

The lengths of a 30-60-90 triangle will always be in the ratio of 1:√3:2. This means that if you know the length of one side, you can easily calculate the lengths of the other two sides. For example, if the length of Side A is 3, then the length of Side B will be 3√3 (or approximately 5.196), and the length of Side C will be 6.