Which important propositions have the simplest proofs?
The Pythagorean theorem
The ABC of Pythagoras
A right triangle, two familiar sides - that's all you need to be able to apply the Pythagorean theorem successfully. Granted, in some cases a calculator can be of great help. Basically, almost all tasks involve calculating an unknown side length. You do this by inserting the known quantities into the Pythagorean theorem and calculating the missing number. The question remains: Where do the "known" variables come from? Answer: These must always emerge from the task. With a little practice, typical tasks in this area can therefore be solved quickly.
The simplest explanation of the Pythagorean Theorem
It's about triangles and how you can calculate the length of the different sides of a triangle. Very important: All of this only works with a right triangle - that means that there has to be a right angle somewhere in the triangle. It doesn't matter where. The three sides of the triangle are each given a letter as a name: a, b and c. What can we do with the Pythagorean theorem? Once we know how long two of the three sides are, we can use them to calculate how long the third side is.
So we know that the page nameda is five inches long and the side with the nameb Now we can figure out how long the page is 12 inches longc is.
To do this, however, we have to take a detour. Only with the length ofaandb if we can't do anything, we have to square them - that is, to the power of 2. The sides of the triangle now become three squares, which collide with each other and emergea, b, andc isa², b² andc² become. Now comes the genius of the Pythagorean theorem: We just have toa² andb²add up and get with itc² out! That means that the two squares of a and b are together the same size as the square of c. If we then take the root of c² we get the length of c - and the problem is solved!
What is the Pythagorean theorem - a look into the details
Read more, promise? Then here comes the scientific definition of the Pythagorean theorem - don't worry, the following can be explained much more easily later: In a flat triangle, the sum of the areas of the squares above the legs is equal to the area of the square above the hypotenuse. Take a short breath and go on.
In principle, this sentence contains all the essential information. However, some of the statements are hidden in such fine details that a thorough "disassembly" is advisable.
First a very important observation: The Pythagorean theorem is only valid in right triangles. Before each calculation, it must therefore be clear that this requirement has been met. Additional remark: Strictly speaking, it must mean that the theorem is only valid in plane right triangles. In general, however, only the geometry of the plane is dealt with in schools. In this respect, this addition can be neglected. For applications in three-dimensional space, however, it is relevant that triangles behave differently on spherical surfaces than non-curved ones.
Now to the sides of the triangle, which have a special meaning for the Pythagorean theorem. Basically every triangle has three sides, which are usually designated with "a", "b" and "c". In the right-angled triangle two of them are so-called cathets, the other is the hypotenuse. How are these differentiated? In the standard designation they are given the names "a" and "b". Opposite the right angle is the hypotenuse "c". It is also characterized by being the longest side in the triangle. However, this cannot always be clearly seen with the naked eye.
The cathetus, hypotenuse and prerequisites for the sentence are thus explained. Now you just need the important knowledge about the ratio of the areas of the squares over the sides. What is actually meant by that?
A square has four right angles and four sides, all of which are the same length. That means: if only one side is known, a square can be drawn from it. For example, the sides a, b and c of the right triangle can serve as the base side. Then there is a structure of a central triangle with three squares that lie directly on the sides.
Why the whole thing? Because the areas of these squares are in a certain ratio to each other: If you add the area of the square over a with that of the square over b, the result is always equal to the area of the square over c. Or more simply: a² + b² = c². And this is the formula for the Pythagorean theorem.
When is the Pythagorean theorem used?
As already mentioned, the Pythagorean theorem is only used for right triangles. However, these are so common in geometry that it is worthwhile to study them intensively. It is also often the case that you can break down complicated surfaces into simpler basic shapes. Even the simple floor plan of a room is often not square or rectangular, but rather composed of several parts. In many cases, right triangles appear.
Calculate with the Pythagorean theorem
In school, the concrete application of the sentence is mostly to calculate missing pages. If you rearrange the formula, you can find the length of any side - as long as you know the other two. Then the following applies:
a² = c² - b²
b² = c² - a²
c² = a² + b²
However, be careful with the solution! As you will see later in the proof, the theorem only works with squares. The claim a + b = c is therefore wrong! In order to calculate correctly with given numbers, you must therefore inevitably be able to deal with square numbers. As a reminder: a² = a ⋅ a.
After the multitude of letters and formulas, now finally an example with numbers. The sides a = 3 centimeters and c = 5 centimeters of a right triangle are known. How big is side b? To do this, you have to use the converted formula:
b² = c² - a²
Then insert the numbers:
b² = 5² cm² - 3² cm² = 25 cm² - 9 cm² = 16 cm²
You now know that b² is 16. However, what you are looking for is not the area of the square, but the length of the side. In plain language, we have to "remove" the square symbol. In order to determine the original number from a square number, the root is calculated. In the example it looks like this:
√b² = √16 cm²
b = 4 cm
The result is: The desired side length is four centimeters.
The second application of the Pythagorean theorem
In addition to calculating the sides of a triangle, there is a second possible application for the set, which is used less often. The reason for this is that the calculation is shorter and simpler - so this area would not fill any classwork. You saw that we can use the Pythagorean theorem to calculate missing sides in a right triangle. You also learned that the theorem only applies to a right triangle. If all sides of a triangle are now known, you can use the sentence to check whether it is right-angled. If this is the case, a² + b² = c² must apply.
You can test this, for example, on the triangle a = 5 centimeters, b = 12 centimeters and c = 13 centimeters.
a² + b² = c²
5² cm² + 12² cm² = 13² cm²
25 cm² + 144 cm² = 169 cm²
169 cm² = 169 cm²
The result is correct, the triangle is therefore right-angled.
The situation is different with the combination a = 1 centimeter, b = 2 centimeters and c = 3 centimeters:
1² cm² + 2² cm² = 3² cm²
1 cm² + 4 cm² = 9 cm²
5 cm² = 9 cm²
This statement is wrong, the corresponding triangle is not right angled!
Additional knowledge: The Pythagorean triple
Three natural numbers, which, as in the first example, fulfill the Pythagorean theorem, are considered special in mathematics. They are called the Pythagorean triples. Historical finds show that people knew the meaning of such triples thousands of years ago.
Why does the Pythagorean theorem apply? How can you prove it?
The fact that the Pythagorean theorem is named after the Greek philosopher Pythagoras of Samos today has to do with the proof. He is said to be the first person to have found evidence of general validity. Whether he was actually the first, however, is controversial. The fact is that to this day he is by no means the only one. There are now over 100 different pieces of evidence. Even this abundance shows that the Pythagorean theorem is very important for geometry.
Much of the evidence is very mathematical and quite creative. Fortunately, among the numerous variants there are also some examples that are simple and understandable. In the following, an algebraic proof (by calculating with unknowns) and a geometric proof (by considering triangles and squares) are presented. Depending on the type of learner, you can choose which method you prefer - both approaches lead to the same goal.
Before that, however, you can see a very practical proof of the Pythagorean theorem here. Here someone has taken the trouble to recreate the squares that adjoin the right-angled triangle as Plexiglas containers and connect them to one another. A colored liquid was then filled into the containers. As you can see here very nicely, the liquid in the two small squares - ie a² + b² - fills exactly the large square, ie c². So the Pythagorean theorem is correct!
Algebraic proof for the Pythagorean theorem
This proof cannot be made entirely without geometry. What is needed is a square with side length a + b, which accordingly has the area (a + b) ². In each corner of this square you can draw a right-angled triangle, the legs of which are a and b. A segment c in the interior of the quadrilateral results as a hypotenuse. In total there are four such triangles, the hypotenuses of which meet at right angles and in turn enclose a square with the area c². Together with the areas of the triangles (four times a ⋅ b / 2 = 2⋅a⋅b) they fill the entire contents of the large square. As an equation it follows:
(a + b) ² = 2⋅a⋅b + c² (two ways of representing the area, which of course have to give the same result)
The bracket on the left is solved with the help of the first binomial formula:
a² + 2⋅a⋅b + b² = 2⋅a⋅b + c²
Because the term 2⋅a⋅b appears on each page, you can shorten it out. What remains is the well-known form of the Pythagorean theorem:
a² + b² = c²
For the geometric proof, two squares with side lengths a + b are required.
The first square is divided into four right triangles and a square with side length c, analogous to the algebraic proof. The second square is divided so that there are two smaller squares with sides a and b respectively. What remains are two rectangles with side lengths a and b. You can put two right-angled triangles in each, which exactly correspond to those from the first square. Both squares therefore contain four equal triangles, which therefore have the same area. The remaining area of the square corresponds to c² once and a² + b² once. Consequently, these must also be of the same size - here, too, the Pythagorean theorem results.
What does the Pythagorean theorem bring? Why do I have to learn it?
Formulas, calculation examples, a Greek philosopher and a lot of triangles are now behind you. It goes without saying that the question now arises: Why all that? There are several good reasons for studying Pythagorean theorem in school. First of all, it is an important part of the geometry. You will come across right-angled triangles a lot during your school days. With the sentence you have a good tool at hand to calculate it. In addition, the calculation of side lengths in right-angled triangles is often used to work on other subject areas. Powers and roots are closely linked to the Pythagorean theorem. It shows in each case what you need the individual competencies for.
If you want to dig deeper into math problems, you will increasingly have to work math. This basically includes conclusive evidence. The Pythagorean Theorem is a very beautiful historical example of this. Not only that it has set a monument to its namesake, it is also comparatively manageable and intuitively understandable. In this way you will learn what a mathematical proof can look like and how it is derived.
Learning the Pythagorean theorem is easy as it only contains the first three letters of the alphabet. Looking up formulas of this type takes too much time in most cases. A good tip to be successful in class work on the Pythagorean Theorem is therefore: learn a² + b² = c² by heart!
Can I use the Pythagorean theorem in everyday life?
During the recent storm, a large beech tree fell over at the end of your street. The fracture point is two meters high, the tip is ten meters further on the earth. Everyone is puzzling: How big was the stately tree actually? By chance you just learned about the Pythagorean Theorem in school ...
Admittedly, you know this situation more from textbooks than from real life. Nevertheless, the Pythagorean theorem is by no means unimportant in everyday life. It is often based on calculations that you use frequently. For example, it is used in 3D graphics, where distances in three-dimensional space are calculated in this way. Fortunately, smartphones and the like take over the often complex bills. This means that you do not have to determine the position of the ball in the football simulation or the distance to the dragon in the adventure game yourself.
In real daily life, the second benefit of the sentence is much more relevant. Knot cords were an important aid even in ancient times. They were used to represent right angles. The twelve-knot cord, which may have been used to build the pyramids, is particularly well-known.
The principle of operation is as follows: The cord is divided into twelve equally long sections with the help of twelve knots. If you make a triangle with sides three, four and five "knots", the result is a right angle. The number triplet may already look familiar to you from an example above - it is the simplest Pythagorean triple.
Craftsmen and especially bushcrafters and hobbyists still use this method to determine right angles.
In the end, however, one thing remains important: The main thing is that you learn to learn. As trite as this sentence is, it is also correct. If you manage to deal successfully with the Pythagorean theorem in school and to understand and apply it, then these experiences will bring you a lot later. What you learn in school is not always useful in everyday life - even if that should be annoying you right now.
In which class is the Pythagorean theorem learned?
Whether it is the eighth, ninth or tenth grade in which you come across the sentence for the first time depends on the state and the school you attend. In one of the grade levels, the famous formula will definitely be the subject.
In any case, you need the Pythagorean Theorem as a basis for almost all of the following classes. Even in high school it is occasionally necessary to use it to calculate a right triangle.
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