Equal Area and Perimeter

Let point $P$ be at the center of the first four circles in the square cluster, and point $Q$ be at the center of the 5th circle. Then the line $PQ$ passing through both will divide both the total area and total perimeter by exactly half.

To understand why, first note that the line PQ runs through the center of the 5th circle, dividing that circle's perimeter and area in half. Now, for symmetry reasons, because line PQ runs through the point that is at the center of the square cluster of circles, however it divides the circles on one side, it divides the circles on the other side equally, thus dividing the total area and total perimeter of those four circles in half. Hence, the total area and total perimeter of all five circles are both divided in half.

Here's another way [per Chung Kevin] to split the five circles. First, add a sixth circle (green). Let point $P$ be at the center of the six circles, and point $Q$ be at the center of the 6th (green) circle. Then the line $PQ$ passing through both will divide both the total area and total perimeter by exactly half not only of all six circles, but of the five black circles as well.

The reasoning is the same as above, with the difference is that since the sixth circle (green) is split in half, area and perimeter, then we can subtract this circle from the cluster of six circles, leaving the line splitting the remaining five circles in half as well, area and perimeter.

Fill in the missing circle, shown in dotted line.

Take the diagonal line that connects the center of the upper right (dotted) circle, and the center of the 6 circles (tangent point of the 2 middle circles).
This line would perfectly split the area and the perimeter of all 6 circles, and also of the 1 dotted circle.
Thus, this line would perfectly split the area and the perimeter of the 5 remaining circles.

If you take the radius of the circle to be 1 unit, and the origin as the center of the downleft circle, the center of mass (for both area and perimeter) would be $\frac{4(1,1) + (4,0)}{5}=(8/5,4/5)$ , and every line going to that point is a valid answer.