717 Symmetrical Arcs and a Line | Centroid of Composite Line

$L_1 = 2r\alpha = 2(30)\left( 30^\circ \times \dfrac{\pi}{180^\circ} \right)$

$L_1 = 10\pi = 31.42 \, \text{ cm}$
 

$z = \dfrac{r \sin \alpha}{\alpha} = \dfrac{30 \sin 30^\circ}{30^\circ (\pi / 180^\circ)}$

$z = 28.65 \, \text{ cm}$
 

 

$y = z \sin \alpha = 28.65 \sin 30^\circ$

$y = 14.325 \, \text{ cm}$
 

$x = 10 + r - z \cos \alpha = 10 + 30 - 28.65 \cos 30^\circ$

$x = 15.19 \, \text{ cm}$
 

$L = 20 + 2L_1 = 20 + 2(31.42)$

$L = 82.84 \, \text{ cm}$
 

By symmetry
$\bar{x} = 0$           answer
 

$L \, \bar{y} = \Sigma ly$

$82.84\bar{y} = 31.42(14.325)(2) + 20(0)$

$\bar{y} = 10.87 \, \text{ cm}$           answer