Every student knows that the square of the hypotenuse is always the sum of the legs, each of which is squared. This statement is called the Pythagorean theorem. It is one of the most famous theorems of trigonometry and mathematics in general. Let's consider it in more detail.
The concept of a right triangle
Before proceeding to the consideration of the Pythagorean theorem, in which the square of the hypotenuse is the sum of the legs that are squared, we should consider the concept and properties of a right triangle, for which the theorem holds.
A triangle is a flat figure having three angles and three sides. A right-angled triangle, as its name implies, has one right angle, that is, this angle is 90 o .
From the common properties for all triangles, it is known that the sum of all three angles of this figure is 180 o , which means that for a rectangular triangle, the sum of two angles that are not straight is 180 o - 90 o = 90 o . The latter fact means that any angle in a right triangle that is not straight will always be less than 90 o .
The side that lies against a right angle is called hypotenuse. The two other sides are the legs of the triangle, they can be equal to each other, and can differ. From trigonometry it is known that the greater the angle against which the side in the triangle lies, the greater the length of this side. This means that in a right triangle the hypotenuse (lies against an angle of 90 ° ) will always be larger than any of the legs (lie against angles <90 ° ).
Mathematical notation of Pythagoras theorem
This theorem says that the square of the hypotenuse is the sum of the legs, each of which is previously squared. To mathematically write down this formulation, we consider a right-angled triangle in which the sides a, b, and c are two legs and a hypotenuse, respectively. In this case, the theorem, which is formulated as the square of the hypotenuse is equal to the sum of the squares of the legs, the following formula can be represented: c 2 = a 2 + b 2 . From here, other important formulas for practice can be obtained: a = √ (c 2 - a 2 ), b = √ (c 2 - a 2 ) and c = √ (a 2 + b 2 ).
Note that in the case of a right-angled equilateral triangle, that is, a = b, the wording: the square of the hypotenuse is the sum of the legs, each of which is squared, is mathematically written as: c 2 = a 2 + b 2 = 2a 2 , which implies the equality: c = a√2.
History reference
The Pythagorean theorem, which states that the square of the hypotenuse is the sum of the legs, each of which is squared, was known long before the famous Greek philosopher paid attention to it. Many papyri of ancient Egypt, as well as clay tablets of the Babylonians confirm that these peoples used the marked property of the sides of a right triangle. For example, one of the first Egyptian pyramids, the Chephren pyramid, the construction of which dates back to the XXVI century BC (2000 years before the life of Pythagoras), was built on the basis of knowledge of the aspect ratio in a 3x4x5 right-angled triangle.
Why, then, is the theorem now named after the Greek? The answer is simple: Pythagoras is the first to mathematically prove this theorem. The preserved Babylonian and Egyptian written sources speak only of its use, but no mathematical proof is given.
It is believed that Pythagoras proved the theorem in question by using the properties of similar triangles, which he obtained by drawing a height in a right-angled triangle from an angle of 90 o to the hypotenuse.
An example of using the Pythagorean theorem
Consider a simple task: it is necessary to determine the length of the inclined staircase L, if it is known that it has a height H = 3 meters, and the distance from the wall into which the staircase rests to its foot is P = 2.5 meters.
In this case, H and P are legs, and L is the hypotenuse. Since the length of the hypotenuse is equal to the sum of the squares of the legs, we get: L 2 = H 2 + P 2 , whence L = √ (H 2 + P 2 ) = √ (3 2 + 2,5 2 ) = 3,905 meters or 3 m and 90, 5 cm