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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.

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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 Problems

Let’s take a look at some examples.

Example 1

This is a 30-60-90 triangle with one side length given. Let’s find the length of the other two sides, a and b.

example problem 1

Since the side you are given, 8, is across from the 30 degree angle, it will be the shorter leg. To find the longer leg, or a, you can simply multiply it by the square root of 3 to get 8 square root 3. To find the hypotenuse, or b, you can simply multiply by the shorter leg by 2. Thus, it will be 8 * 2 = 16.

Example 2

Here is a 30-60-90 triangle with one side length given. Let’s find the length of the other two sides, x and y.

example problem 2

You are given the length of the hypotenuse in this problem. Therefore, you must first find the length of the shorter leg, which is x. You can do this by dividing the hypotenuse, 20, by 2 to get x = 10. Now that you know the value of the shorter leg, you can multiply it by the square root of 3 to find the y.

The longer leg will be 10 square root 3.

Example 3

Here is a 30-60-90 triangle with one side length given. Let’s find the length of the other two sides, c and d.

example problem 3

The side length you are given here, 9, is the value of the longer leg since it’s across from the 60 degree angle. Thus, you must first find the value of the shorter leg, c, before you can determine the value of the hypotenuse, d. To find c, you will need to divide 9 by the square root of 3.

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 rationalize the denominator by multiplying both the numerator and denominator by square root of 3. Remember that when multiplying and dividing radicals, only the numbers outside of the radical and the numbers inside of the radical can be combined.

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 c.

The full work is shown here:

simplifying 9 over square root 3

Then you will take that value and multiply it by 2 to find the value of d, the hypotenuse. This gives 3 square root 3 * 2 or 6 square root 3.

Lesson Summary

Triangles can be grouped by both their angle measurement and/or their side lengths.

Right triangles are one particular group of triangles and one specific kind of right triangle is a 30-60-90 right triangle. As the name suggests, the three angles in the triangle are 30, 60, and 90 degrees. As a result, the lengths of the sides in a 30-60-90 have special relationships between them that allow you to determine all three when you are only given one.

The hypotenuse is equal to 2 times the length of the shorter leg and the longer leg is equal to the square root of 3 times the length of the shorter leg. These relationships also work in reverse and you can instead divide by 2 and the square root of 3 when needed. Knowing these relationships is important since 30-60-90 triangles are quite common, not only in geometry, but in other areas of math as well.

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