The Pythagorean theorem is a mathematical principle that describes the relationship between the sides of a right triangle. It states that in a right triangle, the sum of the squares of the lengths of the two shorter sides (known as the legs) is equal to the square of the length of the longest side (known as the hypotenuse).

In mathematical notation, the theorem can be written as: \( a^2 + b^2 = c^2 \) where "\(a\)" and "\(b\)" represent the lengths of the legs, and "\(c\)" represents the length of the hypotenuse.

The theorem is named after the ancient Greek mathematician Pythagoras, who is credited with its discovery. It is a fundamental concept in mathematics and has numerous applications in fields such as engineering, physics, and architecture. It is used to calculate the distance between two points in a coordinate system, to determine the height of a building or tree, and to design structures that require stable right angles.

**Examples:**

1. Find the length of the hypotenuse in a right triangle with legs of length 3 and 4.

Answer:

\(a=3; b=4\)

\(c = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = 5\)

2. A ladder is leaning against a wall with the base of the ladder 6 feet away from the wall. If the ladder is 8 feet long, what is the height of the ladder where it touches the wall?

Answer:

\(a=6; c=8\)

\(b = \sqrt{c^2 - a^2} = \sqrt{8^2 - 6^2} = \sqrt{28}\)

3. A square has a diagonal of length 10 units. What is the length of each side of the square? Answer: First, we know that the diagonal of the square is 10 units, so we can set up an equation using the Pythagorean theorem, we get the length of each side of the square:

\(a^2+a^2=10^2\) (\(a\) kvadratın tərəfləri, 10 – diaqonal (burada hipotenuz ))

\(a^2=50\)

\(a=\sqrt{50}=5\sqrt{2}\)

4. A square has an area of 25 square units. What is the length of the diagonal of the square?

Answer: First, we know that the area of the square is 25 square units, so we can set up an equation to solve for the length of each side of the square:

\(S=a^2 =25; a=5; \text{S-area, a-sides} \).

So the length of each side of the square is 5 units. Now we can use the Pythagorean theorem to find the length of the diagonal. In a square, the diagonal is the hypotenuse of a right triangle with sides of length s.

So we have:

\(d^2=a^2+a^2=2a^2\)

\(d^2=2\cdot 5^2=50\)

\(d=5\sqrt{2}\)