Understanding the Pythagorean Theorem

The Teorema di Pitagora, also known as the Pythagorean Theorem, is a cornerstone of geometry, particularly for right-angled triangles. This page provides a comprehensive overview of the theorem, its components, and its formulas.

The diagram on the page illustrates a right-angled triangle with its components clearly labeled. The sides of the triangle are identified as follows:

• C₁: cateto 1 (first cathetus)
• C₂: cateto 2 (second cathetus)
• i: ipotenusa (hypotenuse)

Definition: The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides (catheti).

Highlight: The theorem is expressed mathematically as: A₁ + A₂ = A₃, where A₁ and A₂ represent the areas of squares constructed on the catheti, and A₃ represents the area of the square constructed on the hypotenuse.

The page presents several formulas derived from the Pythagorean Theorem:

1. C₁² + C₂² = i²
2. i = √(C₁² + C₂²)
3. C₁ = √(i² - C₂²)
4. C₂ = √(i² - C₁²)

Example: To find the length of the hypotenuse when C₁ = 3 and C₂ = 4, you would use the formula i = √(3² + 4²) = √(9 + 16) = √25 = 5.

The page also includes additional geometric formulas related to the right-angled triangle:

• Area of the triangle: A = (C₁ × C₂) / 2
• Height of the triangle: h = (C₁ × C₂) / i

Vocabulary:

• Cateto: In Italian, this refers to one of the sides adjacent to the right angle in a right-angled triangle.
• Ipotenusa: The Italian term for hypotenuse, which is the longest side of a right-angled triangle, opposite the right angle.

This comprehensive presentation of the Teorema di Pitagora provides students with a spiegazione semplice (simple explanation) of this fundamental geometric principle, making it an excellent resource for scuola media (middle school) students studying geometry.