Problem 511

We begin by splitting $ABCD$ into three equilateral triangles, as shown. (Confirm for yourself that this is the only way to do this.) [[IMAGE0]] 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 $\text{840}$ { $\text{cm 5 =168}$
cm}$. Since$AB$is a side of an equilateral triangle, then$AB=168 $\text{}$
cm}$.