What is the Pythagorean theorem?

The **Pythagorean theorem** is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. It states that: > In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, if a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse, the theorem can be expressed as: \[ c^2 = a^2 + b^2 \] ### Explanation: This means that if you know the lengths of any two sides of a right triangle, you can calculate the length of the third side. The hypotenuse is always the longest side since it is opposite the right angle. ### Proof (using a geometric approach): One classic proof involves creating squares on each side of the triangle: 1. Draw a right-angled triangle with sides a, b, and hypotenuse c. 2. Construct a square on each of these three sides, so you have three squares with areas \( a^2 \), \( b^2 \), and \( c^2 \). 3. By rearranging or decomposing these squares (there are many visual proofs, such as the rearrangement proof by President Garfield or the one attributed to Euclid), you can show that the combined area of the two smaller squares (on sides a and b) exactly equals the area of the largest square (on side c). This theorem is widely used in various fields including architecture, engineering, physics, and computer graphics, as it provides a straightforward way to relate distances in right-angled triangles. If you'd like, I can also provide specific algebraic or visual proofs to deepen your understanding!