Imagine a shelf. Normally, to support it, you need a support that has three parts — one that is vertically next to the wall, another horizontally next to the shelf and a third diagonally.
This object is a great example of a right triangle and an application of the famous Pythagorean theorem.
“As the shelf has a 90º angle, this iron [na diagonal] supports the shelf. The heavier the material you put on the shelf, the stronger this iron will have to be. This little iron is playing the role of the hypotenuse of a triangle”, says Wladimir Neves, director of the Institute of Mathematics at UFRJ (Federal University of Rio de Janeiro).
The conversation about the Pythagorean theorem came to light recently with the publication of a tweet by digital influencer Felipe Neto. Last Saturday (12), he wrote: “34 years and I still haven’t needed to use the Pythagorean theorem. I think (sic) one day I will… they can’t have forced me to memorize this for nothing (sic)”.
Perhaps, like Neto, many people have never applied the theorem in practical life, but it was certainly present in some aspect of their daily lives.
First of all, a brief review might be helpful. Pythagoras of Samos, a Greek philosopher and mathematician, lived around 570 BC to 495 BC in Ancient Greece. The theorem is named after him, but there are already indications that he was not necessarily the one who coined the formula.
“We have a perspective that this philosopher brought contributions to the development of this theorem, but that in fact there was a diffusion at the time [dessa teoria] geometry”, says Paulo Trentin, professor in the physics department at Centro Universitário FEI.
The theorem involves a simple formula, often memorized to the point of exhaustion by elementary school students: the sum of the squares of the legs is equal to the square of the hypotenuse in a right triangle.
This type of triangle necessarily has an angle of 90º, also called a right angle. The hypotenuse is diagonally opposite the right angle, and the legs are the other two straight sides.
The formula was widely used in Ancient Greece for various purposes, such as measuring land. “It’s a geometric contribution that helped a lot to work on these issues of development of irrigation and planting regions”, exemplifies Trentin.
Currently, it is also constantly referenced in civil construction, even if indirectly. This is an area where most likely the theorem can be seen in your everyday life.
Think about the walls of your house. Imagine that you want to find out if they are really straight. One way would be with the theorem. According to Neves, it is possible to discover this by knowing the value of the hypotenuse. In this case, you would need the height of the wall and also mark a point on the floor where the distance from the wall to the floor is known.
“You take a string and put it on top of the wall and even down [no ponto que você marcou no chão] and you know with the Pythagorean Theorem how much that distance [que seria a hipotenusa] have to,” explains Neves.
A practical example: a wall that is three meters high and the point marked on the floor is at a distance of four meters.
“If you nail a nail on top of that wall and run a string down to the floor, four meters away from the wall, that string must be five meters long. If it’s longer than five meters, your wall is hanging out. Now if it is less than five meters, its wall is slumped inwards”, completes Neves.
Trentin also says that “the use that [os pedreiros] make of the Pythagorean Theorem is impressive, [mesmo que não seja] exactly the traditional theorem”.
A ladder, for example, is another case in which the formula has practical application, according to the physics professor. In this case, all the steps are basically models of right triangles — the height of the step is one leg, the base where the foot is placed is another and the hypotenuse would be the value of the diagonal opposite them.
“The part where you will step on the stairs has to be exactly a region that is very proportional to the height [do degrau] that this triangle has, projected with respect to the hypotenuse, so that you have a slope of a good stride. a ladder they [os construtores] call it a light step”, explains Trentin.
The theorem also gained a lot of repercussion in navigation, especially in the ancient world where the crew needed to locate themselves without the use of technological devices.
“You notice that all navigation was formerly done […] with you measuring angles and calculating the hypotenuse of triangles for you to locate yourself in the sea”, says Neves.
In addition to these practical applications, the theorem is important to understand how mathematics itself works, explains Neves. “Once a theorem is proved, it is proved once and for all, and that is a question […] of the development of mathematics”.
Understanding the Pythagorean theorem involves learning the reasoning of how mathematics itself is constructed — an area of knowledge that uses demonstrations to verify if a rule is correct without applying empirical experiments that may change the result in the future.
Thus, just as the theorem will not change in the future because it has already been demonstrated in different ways, more “basic” operations, such as addition and subtraction, will also remain unchanged, as they all represent the model of thinking about mathematics. Learning this, in the last case, is important for people to be able to have a greater involvement with this discipline, going beyond the idea that it is important only to memorize formulas.
beyond the bills
Dilemmas like those exposed by Neto are not uncommon in the world of mathematics — and teachers themselves know that.
“One thing that is very distant when we are teaching in high school, or even in elementary school, is to bring the importance of this reasoning [lógico] through the calculations that we are doing”, says Neves.
The difficulty of showing that mathematics depends a lot on solving an exercise is also pointed out by Trentin, who, in addition to being a university professor, has already taught in basic education. “My classroom practice with children [aprendei a] is to show them a situation that could refer to the use of this theorem”, he says.
The teacher also states that it is important to note that mathematics is not necessarily like that seen in books. The logic that a bricklayer uses to think of a ladder, for example, completely echoes the idea of the theorem, even if the builder doesn’t even remember this mathematical formula.
This, in itself, shows how mathematics is completely applicable in everyday life and, at times, passes unnoticed — ranging from a simple calculation for the payment of a purchase in a supermarket to a logical reasoning that allows the construction of a building properly.
