Pythagorean Theorem – Definition, Formula And Examples

You can’t go out because you have to study the Pythagorean Theorem. What a bad luck! At times like these, you feel like the math is ready. Your head is spinning from so much thinking. Your parents already told you. You have to understand everything about that Theorem or your month’s pay will be suspended. What you gonna do? To study, you have to pass the exam.

The only thing you’re sure of is that you don’t know anything about that theorem. What happened? Well, you didn’t pay any attention in class and the books aren’t helping you at all. And you definitely can’t take the teacher home with you. Study only the theorem? No way! You have to find a solution.

What can you do? You need a simpler explanation to understand everything.  Sometimes the explanations are so complicated that they make this task very difficult. The more they explain to you, the more difficult it is for you to understand. And therefore, you like the subject less.

To know what the Pythagorean Theorem consists of, it is best to have a clear idea of ​​its usefulness. And what is the Pythagorean Theorem useful for? Its applications are multiple. If not, ask engineers, architects, surveyors and even artists.

The mathematician ES Loomis dedicated himself to the study of the proofs of the theorem over time. And, in 1927, he found as many as 350 shows performed. Among those who did, we have Euclid (mathematician), Plato (philosopher). Even the well-known artist and inventor Leonardo da Vinci (a real polymath) took great interest in making these demonstrations. If all of them paid attention to him, it is because the theorem is very useful. Don’t you think?

You have no other way out. You have to learn everything you can about the Pythagorean Theorem. So, let go of laziness and pay attention. Here we will explain everything about this theorem. We will know something about Pythagoras. We will explain why the theorem gets its name. Then we will study its formula. We will introduce you to some of its daily uses. Of course, we will carry out exercises that clarify the dark panorama for you. And between point and point, we have to refer to the triangles, which are the backbone of the theorem. Hello! Let’s go to study.

Instructions

If you want to understand the Pythagorean Theorem, you must be clear about some facts and terms:

• You are probably wondering what a theorem is. A theorem is a statement that you can prove by applying axioms or other theorems. Axioms are also propositions.
• Why is it called the Pythagorean Theorem? Because it was a Greek philosopher, who was also a mathematician, named Pythagoras of Samos, who verified it in the 6th century BC Let’s make this clear, he did not create it. However, both he and the Pythagorean school were credited with being the first to formally demonstrate it.
• So who created it? Some say that he was in India and Babylon; others, in Egypt and Mesopotamia. Authorship cannot be given with certainty to any particular culture.
• And what is? The Pythagorean Theorem is a truth (hence, provable), which is useful only with right triangles. So don’t try to use it with other types of triangles.
• A triangle is, as you already know, a geometric shape that is characterized by having three sides; and of course, three angles. There are various types of triangles, each with its name.
• A right triangle, which is the one we are interested in here, has the property of having an angle of 90°. This angle is called “right”. And, of course, that’s why it’s called a right triangle.
• In these types of triangles, the longest side is called the “hypotenuse.” The sides that meet form the 90° angle are called “legs”. Summing up, two legs and a hypotenuse.
• What is the Pythagorean Theorem useful for? It will let you know the square of the hypotenuse. How? Well, by adding the squares of each of the squares of its legs. It sounds like a simple thing, but this theorem was a major breakthrough for mathematics.
• Let’s get it done! The formula of the Pythagorean Theorem is a ² + b ² = h ².  Both a and b are the legs and h, the hypotenuse.
• With this formula, if we know two data, we will have the unknown. For example, if we don’t know the square of the first leg, we use the formula like this h ² – b ² = a².  For the second leg, we also subtract the first from the hypotenuse. Easy!
• Let’s stop a bit. How do you determine the square of a number? Multiplying it by itself. If we have 2², we must multiply 2×2, which will give us 4. Do they remind you of square roots in any way? It is the reverse process. That’s why you must know how to find the square root of a number.
• Surely you are wondering how useful the Pythagorean Theorem is. Believe it or not, it has many applications. It allows us to know measures, so necessary in various areas of knowledge. It is essential for the work of engineers and architects. And not to mention the artists. Galileo Galilei, was not far behind either; with it he measured the height of mountains. As you can see, it has multiple applications.
• The Theorem allows you to calculate distances, the height of a building or the length of the staircase you want to build. What you should always keep in mind is that a right triangle must intervene in all of them. In the case of the height of the building as well? You are right. To do this, you must consider the size of the shadow that it projects. Then, measure the distance between the highest point of said building, to the end of that shadow.
• Throughout history, the proofs of the Pythagorean Theorem have provided us with what is known as “Pythagorean triples ”. Each of them is formed by three natural numbers. These are the lengths of each of the sides of a right triangle; therefore they satisfy this theorem. It is something like that, like working on insurance. Why?  Simple! It has already been proved, therefore the result is known.
• An example of a Pythagorean triple is 3, 4, 5. If we apply the formula we have: 3 ² + 4 ² = 5². Look, the triples are made up of whole numbers.
• Finally, the best thing to learn to use the formula is to have exercises as examples. Remember that practice makes perfect.

• Another exercise?

What do you need:

• The formula of the Pythagorean Theorem. We hope that by now, it is already clear enough for you. If not, read again.
• A calculator, even if it’s the one on your computer. The one on your mobile could also be useful.
• The multiplication tables, to practice them, just in case.
• Information about the Pythagorean triples. They would be very useful when studying.

Tips

• If you won’t be allowed to use your calculator on the test, practice your multiplication skills.
• Study with your classmates. Team learning is very useful. Allows the exchange of ideas.
• However, remember that you will be alone when doing the exam exercises. Do not depend on others to successfully perform the exercises.
• It is always good to ask your parents what they know on the subject. You will be surprised at what you can learn from them.
• Do not forget what you have studied so far, especially concerning the square root.
• Practice and practice to be a master at the Pythagorean Theorem. Do all the exercises that you have been assigned to study, as well as those that have been solved in class.
• Even if you take your lucky charm with you, trust more in what you have learned and practiced.
• The night before the test, kick back and relax. Clearing the mind helps to concentrate better the next day.
• Don’t let your nerves get the better of you. Answer the test calmly and very carefully. A wrong number, and the exercise will not have the result it should.

After the exam, rest and enjoy a well-deserved rest.