$L_1 = 6 \, \text{ in.}$
$x_1 = 4 \, \text{ in.}$
$y_1 = 3 \, \text{ in.}$
$L_2 = \pi(4) = 12.5664 \, \text{ in.}$
$x_2 = 0$
$y_2 = \dfrac{2(4)}{\pi} = 2.5465 \, \text{ in.}$
$L_3 = 8 \, \text{ in.}$
$x_3 = 4 + 4 \cos 30^\circ = 7.4641 \, \text{ in.}$
$y_3 = 4 \sin 30^\circ = 2 \, \text{ in.}$
$L = L_1 + L_2 + L_3$
$L = 6 + 12.5664 + 8$
$L = 26.5664 \, \text{ in.}$
$L \, \bar{x} = \Sigma lx$
$26.5664\bar{x} = 6(-4) - 12.5664(0) + 8(7.4641)$
$\bar{x} = 1.34 \, \text{ in.}$ answer
$L \, \bar{y} = \Sigma ly$
$26.5664\bar{y} = 6(3) + 12.5664(2.5465) + 8(2)$
$\bar{y} = 2.48 \, \text{ in.}$ answer