Introduction
Pythagoras was a Greek mathematician who discovered and demonstrated an interesting property regarding right triangles. This property states that the sum of the squares of both of the triangle’s legs, which are the sides that form the right angle, is equal to the square of the triangle’s hypotenuse, the side that is an exact reverse of the right angle. Pythagoras is credited with this discovery.
Pythagoras was a philosopher and one of the first mathematical theorists. He lived around the 500s B.C. (Gupta, 2021). Pythagoreans had a strong interest in philosophy, music and mathematics. The theorem’s assertion was found written on a Babylonian tablet that dates back to between 1900 and 1600 B.C. The theorem may be expressed algebraically in the following way: a2 + b2 = c2. Numerous applications may be found for this Pythagorean theorem in science, art, technology, and construction.
Using this simple yet effective equation, we may build our confidence in performing operations on numbers using exponents. Additionally, due to the prevalence of right triangles, we will better understand how beneficial it is to grasp exponential notation fully. The nicest aspect is that we are not required to communicate in Greek.
The Pythagorean Theorem may also be proven using the opposite argument. In other words, a triangle is considered a right triangle if and only if it meets Pythagoras’ Theorem. To express this idea differently, right triangles can only satisfy the Pythagorean Theorem.
At this point, it is essential to have a look at the classic approach to proving the formula for Pythagoras’ theorem, which states that the sum of the squares on the two shorter sides is equal to the square on the hypotenuse, which is the longest side of the triangle. The Pythagorean theorem may also be understood because the square produced by the hypotenuse side is equivalent to the squares created by the perpendicular face and the base side.
Pythagoras’s theorem can be understood geometrically by considering a right-angled triangle, where one of the angles is a right angle (90 degrees). The hypotenuse is the side opposite the right angle; the other two are called the legs. If you draw a square on each of the three sides of the right-angled triangle, then the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
Consider the smaller squares on the legs to see why this is true. The area of the square on the shorter leg is equal to the length of that leg squared (a2), and the area of the square on the longer leg is equal to the length of that leg squared (b2). Here, it is reasonable to consider the larger square on the hypotenuse. The hypotenuse length is c, so the area of the square is equal to c2. Since the triangles that make up the three squares are all congruent (they have the same shape and size), one can see that the sum of the areas of the squares on the legs is equal to the area of the square on the hypotenuse.
The converse of the Pythagorean theorem holds as well, asserting that a triangle has right angles if the squares of any two of its sides are equal to the square of the triangle’s third side. People could demonstrate the reality of irrational numbers with the assistance of Pythagoras’ Theorem (Gupta, 2021). The latter has also served as a crucial connection for numerous laws and theorems, each of which has had a varied and broad-reaching influence on the world of mathematics and, by extension, our everyday lives.
Construction and Architectural Practices
The Pythagorean theorem lets one compute the diagonal line length that connects the two straight lines. This application is regularly used in the construction industry, whether for woodworking, building, or other related work (Vedantu, 2023). It seems reasonable to consider the following scenario: an individual is constructing a roof with a slope. They may use the Pythagorean theorem to get the diagonal length of the roof’s slopes if they already understand the rooftop height and the area it will cover. They might calculate the surface space of the roofing that has to be shingled with this data, or they might use it to cut beams of the appropriate size to sustain the roof.
Laying Out Square Angles
The theorem is used in the building industry to ensure that structures are constructed in a square shape. A triangle whose side lengths coincide with the theorem will constantly be a right triangle (Seehorn, n.d.). An example of such a triangle would be one with side lengths of 3 feet, 4 feet, and 5 feet.
When creating a squared corner between two supports or laying down the foundation for a building, construction crews will make a triangle made of three strings that are the appropriate lengths to correlate with these lengths. If the latter ends of the strings were measured accurately, the corner opposite the triangle’s hypotenuse would be a right angle. This will let the builders know they are erecting the walls or structures on the proper lines.
Navigation
The theorem is an essential tool for navigating in a two-dimensional space. One may determine the shortest distance by using it with two lengths (Vedantu, 2023). For instance, if one is out at sea and they are trying to navigate to a spot that is 200 miles north and 500 miles west, one can utilize the theorem to determine the distance from their ship to that spot and figure out the number of degrees to the west of north that they would need to follow to reach that spot. The diagonal of the triangle will be the line with the smallest distance between the two legs of the triangle, which will be the distances to the north and west.
The navigation of aircraft may be accomplished using the same fundamental concepts. For instance, an aircraft may consider both its altitude above the ground and its proximity to the airport at which it is a route to determine where the appropriate point is to start its descent toward the airport.
Surveying
Surveying is the procedure that cartographers go through before generating a map, in which they determine the numerical measurements and heights between various places on the landscape. Surveyors are tasked with devising methodical approaches to distance measuring to account for the often uneven terrain they encounter. The theorem is employed to determine how steep the slopes of hills and mountains are.
A surveyor will aim a telescope toward a measuring stick a predetermined distance away (Seehorn, n.d.). This will ensure that the line of sight from the telescope and the measurement stick will meet at a correct angle. The surveyor might utilize the theorem to calculate the slope length that encompasses the horizontal distance between the telescope and the measuring stick because they already know both the height of this stick and the horizontal distance between the stick and the telescope.
Crime Scene Investigation
Forensic investigators use the Pythagorean theorem to calculate the trajectory of a gunshot. The trajectory of a bullet illustrates the route that the bullet traveled before contact. The trajectory gives the authorities information about the location where the bullet was fired. The detectives might also determine the distance between the gunman and the victim. This information may assist law enforcement in determining whether or not a death was the result of a suicide or a murder (Seehorn, n.d.).
The Pythagorean analysis is another method that may be used to investigate blood spatter data. The spray of blood that comes from a person after they have been assaulted is known as blood spatter. The computations allow the authorities to calculate the angle of impact and the postures held by both the victim and the offender while the attack occurred.
Location of the Earthquake
The Pythagorean theorem is another tool that geologists use while analyzing earthquake activity. The waves generated by earthquakes are distinct, with one kind being much slower than the other (Vedantu, 2023). Geologists can locate the epicenter or cause of an earthquake by performing a triangulation calculation that compares the distance traveled by quicker waves to that traveled by slower waves.
Trajectory of an Arrow or a Missile
Archers use the Pythagorean Theorem to establish the appropriate trajectory that must be followed to reach their target. If the calculations are correct, the arrow will be successful (Seehorn, n.d.). In such a case, there is a chance that the arrow will not reach its target. Guided missile systems use a comparable approach to attack their targets correctly.
Conclusion
Since it was first discovered, the Pythagorean theorem has significantly impacted how people live. It has been crucial in producing many machines and tools, many of which may trace their origins back to it. The simplicity and condensed form of the theorem, with its general applicability in theoretical and practical elements of all facets of human existence, was a game-changer in how people thought and carried out their daily activities. Therefore, throughout history, it has been serving as the foresight of contemporary endeavor, assisting us in developing numerous luxuries that we now take for granted.
References
Gupta, A. (2021). Pythagorean Theorem research paper 2021. Journal of Research in Applied Mathematics, 7(7), 13–18.
Seehorn, A. (n.d.). Real life uses of the Pythagorean Theorem. eHow. Web.
Vedantu. (2023). Application of Pythagorean Theorem in daily life. Web.