# Pythagorean Theorem

This article presents an intuitive approach to understanding the Pythagorean Theorem.

## Introduction

### Triangle Relationships

In previous lessons, you explored some patterns in triangles. For instance, all the angles inside a triangle add up to 180 degrees. It is also the case that similar triangles have equivalent corresponding angles and corresponding side lengths that are proportional to each other in the same ratio. Right-angle triangles are more orderly, which gives rise to more relationships. You will explore the relationship between the side lengths of right triangles, in the following lesson.

### Conventions

The relationship you will be studying is usually called the Pythagorean Theorem (or minor variations thereof), in English and "Western" countries. Sometimes the slang "Pythagoras" is used instead. In China, it is called the Gougu Theorem. Historically, India has called this relationship the Baudhayana Theorem, but has adopted the English naming convention more recently.

There are many conventions for identifying the side lengths of a right-angle triangle when using the Pythagorean Theorem. In Chinese, the hypotenuse is called 弦/Xian and the other two sides are called 勾/Gou and 股/Gu. In English, it is common to use the letters , , and to identify the side lengths. People often switch between using the and the for the hypotenuse and then use the remaining two letters for the other two sides. Identifying the hypotenuse is the most important part. The other two sides are interchangeable. This is because you can simply flip and rotate the triangle, without changing it. This website will choose to follow (wherever possible) the convention that sets the letter as the value of the length of the hypotenuse. The letters and will be used for the other two sides.

### An Experimental Approach

3 | 4 | 5 |

5 | 12 | 13 |

8 | 15 | 17 |

7 | 24 | 25 |

At this point, an ambitious student could begin drawing right-angle triangles and looking for a pattern in the side lengths. The pattern is fairly complex, and the numbers are really messy unless very specific choices are made. To avoid unnecessary frustration, a table with a few sets of side lengths is provided. They are special numbers called Pythagorean Triples. Their special property is that they are all whole numbers and can be used as the lengths of a right-angle triangle (ie. they satisfy the relationship you are looking for). If you'd like to sketch them (to check that they work) feel free. Now please take a moment and try to find the pattern that relates these triples.

It is worthwhile to note that each Pythagorean Triple has a twin Pythagorean Triple, in which the and values have been swapped. If you took , , and to be valued 3, 4, and 5; then another valid triple would be 4, 3, and 5.

It is also valuable to notice that similar triangles sometimes also work. For example, if you were to multiply all the values of a Pythagorean Triple by a single positive whole number, the new values would also form a new Pythagorean Triple. Explicitly, if you took , , and to be valued 3, 4, and 5; then if you were to multiply each of these values by 2 to get 6, 8, and 10; you would find that 6, 8, and 10 make another Pythagorean Triple.

Feel free to take a break if you need it. The pattern is particularly unexpected, so there is no shame in moving on to the next section, once you've made a strong effort.

## Finding the Pythagorean Theorem: A Visual Approach

Now that you have tried to find the relationship, take a look at this app. Leave the action button (beside the full-screen button) and hint buttons alone for now. You can use the full-screen button and slider if you like.

What do you see? What does the slider do?

## Finding the Pythagorean Theorem: An Algebraic Approach

Here is another way to justify the Pythagorean Theorem. It is my favorite. It requires more algebra, but has the benefit that it is more direct. It does not require an elaborate construction and, therefore (in my opinion), is more likely to be understood (or even reinvented) by a first-time learner. Have a look at the diagram provided and see if you can justify the Pythagorean Theorem in a new way.

## Other Derivations

The Pythagorean Theorem can be derived in many ways. I invite you to investigate other approaches. There are many good videos on YouTube and an article on Wikipedia. Feel free to Google other derivations as well.

## Conclusion

The Pythagorean Theorem is an important result in trigonometry. It lays some of the groundwork for learning about the distance formula, vectors, polar coordinates, and even the Euclidean Metric.