This lesson will explore a specific kind of right triangle, the 30-60-90 right triangle, including the relationships that exist between the sides and angles in them.

## Special Triangles

If you’ve had any experience with geometry, you probably know that there are many different types of triangles. They can be classified by side length (isosceles, scalene, or equilateral) or by angle measurement (acute, obtuse, or right). Additionally, some of these types can be classified even further into smaller groups. This lesson is going to examine one kind of **right triangle**, which is a triangle that has exactly one right, or 90 degree, angle.

This specific kind is a **30-60-90 triangle**, which is just a right triangle where the two acute angles are 30 and 60 degrees. Why does this specific triangle have a special name? Let’s find out.

## Qualities of a 30-60-90 Triangle

A 30-60-90 triangle is special because of the relationship of its sides. Hopefully, you remember that the **hypotenuse** in a right triangle is the longest side, which is also directly across from the 90 degree angle.

It turns out that in a 30-60-90 triangle, you can find the measure of any of the three sides, simply by knowing the measure of at least one side in the triangle.The hypotenuse is equal to twice the length of the shorter leg, which is the side across from the 30 degree angle. The longer leg, which is across from the 60 degree angle, is equal to multiplying the shorter leg by the square root of 3. This picture shows this relationship with *x* representing the shorter leg.

Of course, to go in the opposite direction you can divide, instead of multiply, by the appropriate factor. Thus, the relationships can be summarized like this:Shorter leg —> Longer Leg: Multiply by square root of 3Longer leg —> Shorter Leg: Divide by square root of 3Shorter Leg —> Hypotenuse: Multiply by 2Hypotenuse —> Shorter Leg: Divide by 2Notice that the shorter leg serves as a bridge between the other two sides of the triangle. You can get from the longer leg to the hypotenuse, or vice versa, but you first ‘pass through’ the shorter leg by finding its value. In other words, there is no direct route from the longer leg to the hypotenuse, or vice versa.
## Example ProblemsLet’s take a look at some examples. ## Example 1This is a 30-60-90 triangle with one side length given. Let’s find the length of the other two sides,
## Example 2Here is a 30-60-90 triangle with one side length given. Let’s find the length of the other two sides,
The longer leg will be 10 square root 3. ## Example 3Here is a 30-60-90 triangle with one side length given. Let’s find the length of the other two sides,
You should recognize though that once you do this, the expression you get, 9 / square root 3, needs to be simplified since you are not allowed to have a radical in the denominator of a fraction. To simplify it, you will need to The numerator will become 9 square root 3, and the denominator becomes square root 9, or just 3. Thus, you now have (9 square root 3) / 3. The 9 on top and the 3 on the bottom can be canceled out, since they are both outside of the radical, leaving a final answer of 3 square root 3 for The full work is shown here:
## Lesson SummaryTriangles can be grouped by both their angle measurement and/or their side lengths. The x
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