Explain the logic behind the Pythagorean theorem.
The Pythagorean theorem states that in a right 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.

To understand the logic behind the Pythagorean theorem, let's consider a right triangle ABC, where C is the right angle and AB is the hypotenuse. We can draw a square on each of the three sides of the triangle, as shown below:


          A ________ C
           |         / 
           |        /   
           |       /    
           |      /     
           |     /      
           |    /       
           |   /        
           |  /         
           | /          
          B/           

Let a, b, and c be the lengths of sides AB, BC, and AC, respectively. We can label the areas of the three squares as follows:

Area of square on side AB = a^2
Area of square on side BC = b^2
Area of square on side AC = c^2

Since the triangle is right-angled, we know that the two smaller squares on sides AB and BC together form a larger square on side AC. Therefore, the area of the square on side AC is equal to the sum of the areas of the squares on sides AB and BC:

a^2 + b^2 = c^2

This is the Pythagorean theorem in its most basic form. In other words, the Pythagorean theorem is a geometric expression of the relationship between the sides of a right triangle.