As an application of connectedness you can prove the "Borsuk-Ulam" theorem in dimension 1, i.e. that for any continuous function $f$ from $S^1$ to the reals there are two radially symmetric points which are mapped to the same point. This is because the function g(v) = f(v) - f(-v) is either constant or has points where it is positive and points where it is negative, therefore it must have a point where it is zero.

As an application of this fact you can show that for any pair of compact regions A and B inside the plane there is one line splitting each region in pieces of equal area (see the book by Kosniowski, A first course in algebraic topology, where this is referred to as a "pancake problem").