Question

What is the co-ordinates of the centroid the shaded area given below?

• A. (96.12, 69.39)
• B. (96.12, 53.89)
• C. (123.75, 53.89)
• D. (123.75, 69.39)

Answer 1

The correct option is D (123.75, 69.39)

Let the centroid be $(x_c,y_c)$
$x_c=\frac{\sum A_i{x}_i}{\sum A_i}\&y_c=\frac{\sum A_i{y}_i}{\sum A_i}$

$\begin{array}{ccc}\text{Area}\left(A_i\right)& X_i& y_i\\ A_1=\frac{1}{2}\times 50\times 250=62.50& \frac{2}{3}\times 250& 100+\frac{50}{3}=116.67\\ A_2=100\times 250=25000& 125& 50\\ A_3=-\frac{\pi }{8}\times {100}^2=-3927& 200& \frac{4\times 50}{3\pi }=21.22\end{array}$
$x_c=\frac{(6250\times \frac{500}{3})+(25000\times 125)-(3927\times 200)}{6250+25000-3927}=123.75$
$y_c=\frac{(6250\times 116.67)+(25000\times 50)-(3927\times 21.22)}{6250+25000-3927}=69.39$