Square Root of a Sum? - Geometric Interpretation

We know that for non-negative numbers  Can we apply the same for the sum? 
In the example below a and b are the legs of a right triangle.  According to the Pythagorean Theorem the hypotenuse is

• Drag to the left the blue point A to rotate the leg AC around point C until the two legs form one segment.
  
• Drag to the left the orange point A to rotate the hypotenuse around point B until it comes to the base of the triangle.
• Compare the lengths  

Irina Boyadzhiev, 6 April 2013, Created with GeoGebra

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