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.

### 3-4-5 Right Triangle

### Solve problems with 3-4-5 right triangles

### Pythagorean Triple

You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

Recognizing special In these lessons, we will study

- the special right triangle called the 3-4-5 triangle.
- how to solve problems involving the 3-4-5 right triangle
- some examples of the Pythagorean Triples

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* = 3*n*
: 4*n* : 5*n*

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 3*n *:
4*n *: 5*n* 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

5

n= 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 3*n *:
4*n *: 5*n* 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

3

n= 3 × 3 = 9

Answer: The length of the side is 9 inches.

3-4-5 is an example of the Pythagorean Triple. It is usually written as (3, 4, 5).

In general, a Pythagorean triple consists of three positive integers such that *a*^{2} + *b*^{2} = *c*^{2}.

Other commonly used Pythagorean Triples are (5, 12, 13), (8, 15, 17) and (7, 24, 25)

Conversely, any triangle that has the Pythagorean Triples as the length of its sides would be a right triangle.

The following video will define and explain the Pythagorean Triples.
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You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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