Pythagorean theorem

The Pythagorean theorem is directly related to the right triangle, the Egyptians and the Babylonians already used, but have not had the formulation and proper mathematical rigor. The history of the Pythagorean Theorem runs through ancient Greece, where the philosopher and also mathematician Pythagoras performed the first demonstration of this theorem.

It is hypothesized that Pythagoras may have observed ancient mosaics that had the geometric forms triangles isosceles rectangles and triangles scalene rectangles to thus conceive the theorem that bears his name. Here is the geometric structure of these two triangles:

• Isosceles Rectangle Triangle:  This triangle has an angle that measures 90 ° , its other two angles are acute, that is, smaller than 90 ° . In relation to the sides, two are congruent having the same measure.
• Triangle Rectangle Scalene: This triangle has an angle that measures 90 ° , its other two angles are acute, that is, smaller than 90 ° . Already the measure of its sides are all different.

The relation between the right triangle and the Pythagorean Theorem

After Pythagoras observed these mosaics, he established important geometric relations, one of them being:

"The area of ​​the frame built on the hypotenuse is equal to the sum of the areas of the squares built on the legs."

To understand this definition more clearly, look at the geometric figure below, in it is represented the hypotenuse and the hinges of a right triangle.

The hypotenuse will always be the side affixed to the 90 ° angle, since the legs are the straight segments that form the 90 ° angle. Now that we know the characteristics of a triangle rectangle, we will represent the relation previously stated using a geometric example that follows the condition of existence of triangle.

Calculation of the triangle rectangle area

Note that the area of ​​the square on the hypotenuse is equal to the sum of the area of ​​the squares built on the legs, that is:

Area of ​​the square on the hypotenuse = sum of squares built on the hinges

5 x 5 = (4 x 4) + (3 x 3) 
25 = 16 + 9 
25 = 25

Pythagorean Theorem Formula

The formula of the Pythagorean Theorem is described by the following phrase:

"In all the triangle rectangle the measure of the hypotenuse squared is equal to the sum of the squares of the measures of the legs".

Here is the geometric and algebraic representation of this theorem:

Geometric representation and formula of Pythagorean Theorem

We can apply the Theorem to find the unknown measure of one of the sides of the right triangle. Follow the example below and see how this can be done:

Example

Find the value for the measure of (x) in the triangle below. 
Answer:  Replace the values ​​for the measure of the hicks in the formula of Pythagoras and obtain the measure of x.