In the center of the square of side 1 shown in the figure is an ant. At one point the ant starts walking until it touches the left side (a), then continues walking until it reaches the bottom side (b), and finally returns to the starting point. Show that, regardless of the path followed by the ant, the distance it travels is greater than the square root of 2.
[asy]
unitsize(2 cm);draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);label("a", (0,0.5), W);
label("b", (0.5,0), S);
dot((0.5,0.5));
[/asy] geometryGeometric Inequalitiessquare