Every student knows that the square of the hypotenuse is always equal to 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. Consider it in more detail.
The concept of a right triangle
Before proceeding to consider the Pythagorean theorem, in which the square of the hypotenuse is equal to the sum of the legs that are squared, we should consider the concept and properties of a right-angled triangle, for which the theorem is valid.
Triangle is a flat figure with three angles and three sides. A right triangle, as its name implies, has one right angle, that is, this angle is 90o.
From the general properties for all triangles, it is known that the sum of all three angles of this figure is 180o, which means that for a right triangle the sum of two angles that are not right, is 180o -90o=90o. The last fact means that any angle in a right triangle that is not a right angle will always be less than 90o.
The side that lies opposite the right angle is called the hypotenuse. The other two sides are the legs of the triangle, they can be equal to each other, or they can differ. It is known from trigonometry that the greater the angle against which a side lies in a triangle, the greater the length of this side. This means that in a right triangle the hypotenuse (lie opposite the angle 90o) will always be greater than any of the legs (lie opposite the angles < 90o).
Mathematical notation of the Pythagorean theorem
This theorem says that the square of the hypotenuse is equal to the sum of the legs, each of which is previously squared. To write this formulation mathematically, consider a right triangle in which the sides a, b, and c are the two legs and the hypotenuse, respectively. In this case, the theorem, which is stated as the square of the hypotenuse is equal to the sum of the squares of the legs, can be represented by the following formula: c2=a2 + b 2. From here, other formulas important for practice can be obtained: a=√(c2 - b2), b=√(c2 - a2) and c=√(a2 + b2).
Note that in the case of a right-angled equilateral triangle, that is, a=b, the formulation: the square of the hypotenuse is equal to the sum of the legs, each of whichsquared, mathematically written as: c2=a2 + b2=2a 2, which implies the equality: c=a√2.
Historical background
The Pythagorean theorem, which says that the square of the hypotenuse is equal to 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 noted property of the sides of a right triangle. For example, one of the first Egyptian pyramids, the pyramid of Khafre, whose construction dates back to the 26th century BC (2000 years before the life of Pythagoras), was built based on the knowledge of the aspect ratio in a 3x4x5 right triangle.
Why then is the theorem now named after a Greek? The answer is simple: Pythagoras is the first to mathematically prove this theorem. Surviving Babylonian and Egyptian writings only mention its use, but do not provide any mathematical proof.
It is believed that Pythagoras proved the theorem under consideration by using the properties of similar triangles, which he obtained by drawing a height in a right triangle from the angle 90o to the hypotenuse.
An example of using the Pythagorean theorem
Let's consider a simple problem: it is necessary to determine the length of an inclined staircase L, if it is known that it has a height H=3meters, and the distance from the wall against which the ladder rests to its foot is P=2.5 meters.
In this case, H and P are the 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: L2=H2 + P2, whence L=√(H2 + P2)=√(32 + 2, 5 2)=3.905 meters or 3 meters and 90.5 cm.
