[i][b]Please look carefully at the applet “Was Pythagoras Wrong?” before exploring this applet.[/b][/i][br][br][br]In that applet we established that there is a sequence of paths that come closer and closer the to the hypotenuse of an isosceles right triangle. Each of the paths in the sequence has the same length; i.e., the sum of the lengths of the two perpendicular legs of the triangle.[br][br]In an isosceles right triangle with hypotenuse of length [i][b]a[/b][/i], the length of each of the other legs is [math]\sqrt{2}/2[/math][b][i]a[br][/i][/b][br]Thus the sum of the lengths of the two perpendicular legs is [math]\sqrt{2}[/math][i][b]a[/b][/i] and the perimeter of the isosceles right triangle is [math]\left(1+\sqrt{2}\right)[/math][i][b]a[/b][/i][br][br]Why does the length of the blue line not change as it becomes more and more jagged?[br] [br]What is the area enclosed by the shape formed?[br][br]What do you think it means for a curve to be smooth?[br][br]If the distance between two points that are normally thought of as being a distance [i][b]a[/b][/i] apart can be as much as [math]\sqrt{2}[/math][i][b]a[/b][/i] apart, what are the implications for other polygons that have such “sawtooth” edges?[br]