*Nola Bridgens*Show bio

Nola has taught elementary school and tutored for four years. She has a bachelor's degree in Elementary Education, a master's degree in Marketing, and is a certified teacher.

Instructor:
*Nola Bridgens*
Show bio

Nola has taught elementary school and tutored for four years. She has a bachelor's degree in Elementary Education, a master's degree in Marketing, and is a certified teacher.

After completing this lesson, you will be able to identify and describe the characteristics of a 3-4-5 right triangle and use the Pythagorean Theorem. You will also be able to discuss situations where 3-4-5 right triangles would be useful.

A **right triangle** is any triangle with one right angle of 90o. There are several kinds of right triangles, but the 3-4-5 right triangle has special characteristics. The measurements of the sides of a 3-4-5 right triangle are in the ratio 3:4:5. A **ratio** is the relationship between numbers that shows proportion. As long as the length of the sides of a triangle are proportionate to 3:4:5, it is a 3-4-5 right triangle.

To find other triangles that are proportionate, you can multiply each of the numbers in the ratio by the same number. For example, if we multiply by 2 we get 6-8-10, and if we multiply by 10 we get 30-40-50. Both of these triangles are 3-4-5 right triangles because they maintained the ratio of 3:4:5.

If you know the length of all the sides of a triangle, you can determine whether it is a 3-4-5 right triangle simply by checking to see if the ratio of the sides is proportionate to 3:4:5. However, if we do not know the measurements of all the sides, we need to calculate them to see if it is a 3-4-5 right triangle.

If you do not know the length of one of the sides of a triangle, don't worry. A famous Greek mathematician named Pythagoras discovered a formula to help us. Named after himself, the Pythagorean Theorem can help us determine the length of an unknown side of a right triangle. The **Pythagorean Theorem** formula is: *a*2 + *b*2 = *c*2 (*c* is also known as the hypotenuse). As long as we know the length of two of the sides of a right triangle, we can use this method. Let's look at the example. We know that side *a* = 6 and side *b* = 8. If we plug the numbers into our formula, we get:

62 +82 = *c*2

62 = 36 and 82 = 64

36 + 64 = 100

We now know that *c*2 = 100.

The square root of 100 is 10, so we know that *c* = 10.

This triangle has the ratio 6:8:10, which is proportionate to 3:4:5, so it is a 3-4-5 right triangle.

Now you might be saying to yourself, ''This formula and information on 3-4-5 right triangles will help me in math class, but when will I ever use this in real life?'' That is a great question, and I have the answer for you.

Let's think about the following problem: You are helping your dad build a doghouse for your dog Spot, and you need to make sure that the corners of the doghouse are 90o angles. You want the doghouse to be 3 feet tall and 4 feet wide, but you need to make sure the corner angle is exactly 90o . You can use what you know about 3-4-5 right triangles and the Pythagorean Theorem to determine how long the hypotenuse needs to be to make sure you have nice 90o angles on the doghouse.

Let's put the Pythagorean Theorem to work for us. We know that *a* = 3 and *b* = 4.

32 + 42 = *c* 2

9 + 16 = 25

*c* 2 = 25

The square root of 25 is 5, so *c* = 5.

You need the hypotenuse of your triangle in the corner of the doghouse to be 5 feet to have a perfect right angle. Your dog Spot will be very happy that his doghouse is nice and square, all because you learned about the Greek mathematician Pythagoras.

We learned that 3-4-5 **right triangles** have one angle that is 90o and sides that are proportionate to the ratio 3:4:5. We also discovered that the **Pythagorean Theorem** helps us find the length of an unknown side of a right triangle and can be helpful in our everyday life.

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