3-4-5 Right Triangles

Recognizing special right triangles in geometry can help you to answer some questions quicker. A special right triangle is a right triangle whose sides are in a particular ratio. You can also use the Pythagorean theorem, but if you can see that it is a special triangle it can save you some calculations.

There are three types of special triangles: 3-4-5 triangles, 45°-45°-90° triangles and 30°-60°-90° triangles. In this lesson we will study 3-4-5 triangles.

A 3-4-5 triangle is right triangle whose lengths are in the ratio of 3:4:5. When you are given the lengths of two sides of a right triangle, check the ratio of the lengths to see if it fits the 3:4:5 ratio.

Side1 : Side2 : Hypotenuse = 3n : 4n : 5n

Example 1:

Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are 6 inches and 8 inches.

Solution:

Step 1: Test the ratio of the lengths to see if it fits the 3n : 4n : 5n ratio.

6 : 8 : ? = 3(2) : 4(2) : ?

Step 2: Yes, it is a 3-4-5 triangle for n = 2.Step 3: Calculate the third side

5n = 5×2 = 10

Answer: The length of the hypotenuse is 10 inches.

Example 2:

Find the length of one side of a right triangle if the length of the hypotenuse is 15 inches and the length of the other side is 12 inches.

Solution:

Step 1: Test the ratio of the lengths to see if it fits the 3n : 4n : 5n ratio.

? : 12 : 15 = ? : 4(3) : 5(3)

Step 2: Yes, it is a 3-4-5 triangle for n = 3

Step 3: Calculate the third side

3n = 3×3 = 9

Answer: The length of the side is 9 inches.

The following video gives another example of the 3-4-5 triangle

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