The Pythagorean Theorem it is one of the most useful theorems. Base in mathematics, geometry, trigonometry, algebra and widely used in everyday life such as construction, navigation, topography, among others.
The Pythagorean Theorem allows you to find the lengths of the sides of a right triangle, and although many triangles are not right, they can all be divided into two right triangles, where the Pythagorean Theorem can be applied.
BASIC CONCEPTS "To understand the Pythagorean theorem"
Triangle:
Geometric figure, in the plane, formed by three sides that meet at vertices. Vertices are written in capital letters and the side opposite the vertex with the same lowercase letter. See figure 1. In the triangles:
- The sum of two of its sides is greater than the other side.
- The sum of the angles of a triangle measures 180º.
Classification of triangles
Depending on the length of the sides, a triangle can be equilateral if it has three equal sides, isosceles if it has two equal sides, or scalene if none of its sides are equal. See figure 2.
A right angle is one that measures 90 °. If the angle is less than 90 ° it is called an “acute angle”. If the angle is greater than 90 ° then it is called an “obtuse angle”. According to the angles, the triangles are classified into:
- Acute angles: if they have the 3 acute angles.
- Rectangles: if they have a right angle and the other two angles are acute.
- Obtusangles: if they have an obtuse angle and the other acute. See figure 3.
Right triangle:
The right triangle is one with a right angle (90 °). Of the three sides of the right triangle, the longest is called the "hypotenuse", the others are called "legs" [1]:
- Hypotenuse: side opposite the right angle in a right triangle. The longer side is called the hypotenuse which is opposite the right angle.
- Legs: it is either of the two smaller sides of a right triangle that makes up the right angle. See figure 4.
Pythagorean Theorem
Statement of the Pythagorean Theorem:
The Pythagorean Theorem states that, for a right triangle, the hypotenuse squared is equal to the sum of the squares of the two legs. [two]. See figure 2.
The Pythagorean theorem It can also be stated as follows: The square built on the hypotenuse of a right triangle has the same area as the sum of the areas of the squares built on the legs. See figure 6.
With the Pythagorean Theorem You can determine the length of either side of a right triangle. In figure 7 are the formulas to find the hypotenuse or some of the legs of the triangle.
Uses of Pythagora's theorem
Construction:
The Pythagorean theorem It is useful in the design and construction of ramps, stairs, diagonal structures, among others, for example, for calculating the length of a sloping roof. Figure 8 shows that for the construction of building columns, trestles and ropes are used that must comply with the Pythagorean Theorem.
Topography:
In topography, the surface or relief of a terrain is represented graphically on a plane. For example, you can calculate the slope of land using a measuring rod of known height and a telescope. A right angle is formed between the line of sight of the telescope and the rod, and once the height of the rod is known, the Pythagorean theorem is used to determine the slope of the terrain. See figure 8.
Triangulation:
It is a method used to determine the location of an object, known two reference points. Triangulation is used in cell phone tracking, in navigation systems, in the detection of a ship in space, among others. See figure 9.
Who was Pythagoras?
Pythagoras was born in Greece In 570 BC, he died in 490 BC. He was a philosopher and mathematician. His philosophy was that each number had a divine meaning, and the combination of the numbers revealed other meanings. Although he did not publish any writing throughout his life, he is known for introducing the theorem that bears his name, useful for the study of triangles. He is considered the first pure mathematician, who developed mathematical studies in geometry and astronomy. [two]. See figure 2.
Drill
To use the Pythagorean Theorem, the first thing to do is identify where the right triangle is formed, which of the sides is the hypotenuse and the legs.
Exercise 1. Determine the value of the hypotenuse for the right triangle in the figure
Solution:
Figure 12 shows the calculation of the hypotenuse of the triangle.
Exercise 2. A pole is required to be supported by a set of three cables, as shown in figure 13. How many meters of cable must be purchased?
Solution
If the cable is considered as the hypotenuse of a right triangle formed between the cable, the pole and the ground, the length of one of the cables is determined using the Pythagorean theorem. Since there are three cables, the length obtained is multiplied by 3 to obtain the total length needed. See figure 14.
Exercise 3. To transport some boxes, from a second floor to the ground floor, you want to purchase an inclined conveyor belt like the one shown in figure 15. How long must the conveyor belt be?
Solution:
Considering the conveyor belt as the hypotenuse of the right triangle formed between the belt, the ground and the wall, in Figure 16 the length of the conveyor belt is calculated.
Exercise 4. A carpenter designs a piece of furniture where books should go, and a 26 ”television. How wide and high should the partition be where the TV will go? See figure 17.
Solution:
The measurement that is used in electronic devices such as telephones, tablets, televisions, among others in the diagonal of the screen. For a 26 ”TV, the screen diagonal is 66,04 cm. Considering the right triangle formed by the diagonal of the screen, and the sides of the television, the Pythagorean theorem can be applied to determine the width of the television. See figure 18.
Conclusions on the Pythagorean Theorem
The Pythagorean Theorem allows you to find the length of the sides of a right triangle, and even for any other triangle, since these can be divided into right triangles.
The Pythagorean Theorem indicates that the square of the hypotenuse of a right triangle is equal to the sum of the square of the legs, being very useful in the study of geometry, trigonometry, and mathematics in general, with wide use in construction, navigation, topography , among many other applications.
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