The answer given is $x=0$ $y=0.557$ I've tried a lot of ways but still can't get the answer. First is I tried to take it into $5$ parts, $2$ quarter circle + $2$ Area of sectors + A Isosceles triangle with side $4.8$ but I can't get the $0.557$ for the $y$ coordinate. My angles are in radians already. 
Here are the formulas 
Draw two radii down to either end of the lower chord, and call the angle subtended by the chord $2\theta$.
We have $\cos \theta=\frac{3}{4.8}=\frac 58$.
The area of the upper sector is $$A_1=\frac 12r^2\times2(\pi-\theta)$$ The position of the centroid of this sector is at a distance $$\bar{x}_1=\frac{2r\sin(\pi-\theta)}{3(\pi-\theta)}$$ above the centre of the circle.
The lower triangle has area $$A_2=\frac 12r^2\sin {2\theta}$$
The position of the centroid of this triangle is at a distance $x_2=2$ below the centre of the circle.
We can now apply Varignon's Principle: $$A_1x_1+A_2(-x_2)=(A_1+A_2)\bar{y}$$
Plugging in the values, the answer follows fairly easily.