Special Right Triangles

Definition: Special right triangles have side lengths with particular proportions, allowing us to find exact trigonometric values. There are two types of special right triangles, which are identified by their angle measures: 45°-45°-90° triangles and 30°-60°-90° triangles.

45°-45°-90° triangles always have a side ratio of 1 : 1: √2 (leg : leg : hypotenuse). Another way we can look at this is, if the sides = x, the hypotenuse = x√2.

30°-60°-90° triangles always have a side ratio of 1 : √3 : 2 (short leg : long leg : hypotenuse). Another way we can look at this is, if the short leg = x, the long leg = x √3 and the hypotenuse = 2x.


Given the length of one side, we can find the other two by setting up a proportion or finding the value of x for a particular triangle.

Example 1 : Find all missing sides:specialright2.pngSolution:

Step 1: Use the given information to find the missing sides. Because this is a 30-60-90 triangle, we know the long leg has a value of x√3. We are told the length of the long leg is 3, so can write the following equation to find x:

x√3 = 3

x = 3/√3 = √3

Step 2: Now that we know x = √3 for this triangle, we can find the short leg, which has a value of x, and the hypotenuse, which has a value of 2x:

Short leg = x = √3

Hypotenuse = 2x = 2√3

This tells us, for a 30-60-90 triangle, when the long leg is 3, the short leg will always be √3 and the hypotenuse will always be 2√3.

Example 2: Find all missing sides:



Step 1: To find the other sides, we can to apply the appropriate ratio to set up a proportion. Because this is a 45-45-90 triangle, the ratio is 1:1:√2. We are also given the hypotenuse, √5. Set up the proportion, where x is the length of both legs:

Step 2: Solve the proportion and rationalize the denominator:

This tells, for a 45-45-90 triangle, when the hypotenuse is √5, the side lengths will always be √10/2.

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