Pythagorean Theorem

WHAT IS THE PYTHAGOREAN THEOREM?

The Pythagorean theorem is a fundamental principle in geometry that relates to right-angled triangles. According to this theorem, the square of the hypotenuse (the longest side of a right-angled triangle) is equal to the sum of the squares of the legs (the two shorter sides). Mathematically, the theorem is written as c² = a² + b², where 'a' and 'b' are the lengths of the legs, and 'c' is the length of the hypotenuse.

DETAILED EXPLANATION

In this theorem, understanding three key elements of a right-angled triangle is essential:

• LEGS (CATETI): These are the two shorter sides of the triangle that together form the right angle. In our example, these are sides 'a' and 'b'.
• HYPOTENUSE: This is the longest side of the triangle and lies opposite the right angle. In our theorem, this is side 'c'.
• RIGHT-ANGLED TRIANGLE: A triangle that has one angle equal to 90° (a right angle).

APPLICABILITY OF THE THEOREM

The Pythagorean theorem allows for the calculation of the length of any side of a right-angled triangle if the lengths of the other two sides are known. For example, if we know the lengths of the legs 'a' and 'b', we can calculate the length of the hypotenuse 'c' using the formula c = sqrt(a² + b²).

EXAMPLE OF THE THEOREM'S USE

For a better understanding, let's use a practical example: Assume we have a right-angled triangle with legs of lengths 3 units and 4 units. We use the Pythagorean theorem to calculate the length of the hypotenuse. According to the formula, we get c² = 3² + 4², which means c² = 9 + 16 = 25. The hypotenuse of this triangle is therefore c = sqrt(25) = 5 units.

CONCLUSION

This theorem is a fundamental tool in geometry that allows for the understanding and solving of problems related to right-angled triangles. Its versatile applicability and simplicity rank it among the most important mathematical principles.