Problem 519

We begin by splitting $ABCD$
into three equilateral triangles, as shown. (Confirm for yourself that
this is the only way to do this.)

Since the equilateral triangle in the middle shares a side with each
of the two other equilateral triangles, then the sides of all three
equilateral triangles are equal in length.

The perimeter of $ABCD$ is made up of
$5$ side lengths of these equilateral
triangles.

Since the perimeter of $ABCD$ is
$840 \text{ cm}$, then each side
length of these equilateral triangles is $\frac{840\textrm { cm}}{5}=168\text{ cm}$.

Since $AB$ is a side of an
equilateral triangle, then $AB=168 \text{ cm}$.