1. **Problem Statement:** Given the equation of semi-circle A as $y = \sqrt{16 - (x - 4)^2}$, find the equations for semi-circles B, C, and D. 2. **Understanding the formula:** The general equation for a semi-circle centered at $(h, k)$ with radius $r$ is: $$y = k + \sqrt{r^2 - (x - h)^2}$$ Since all semi-circles are upper halves and have radius 4, $r = 4$ and $r^2 = 16$. 3. **Given data:** - Semi-circle A center: $(4,0)$, equation: $y = \sqrt{16 - (x - 4)^2}$ - Semi-circle B center: $(12,0)$ - Semi-circle C center: $(8,4)$ - Semi-circle D center: $(16,4)$ 4. **Write equations for B, C, and D:** - For B, center $(12,0)$, radius 4: $$y = \sqrt{16 - (x - 12)^2}$$ - For C, center $(8,4)$, radius 4: $$y = 4 + \sqrt{16 - (x - 8)^2}$$ - For D, center $(16,4)$, radius 4: $$y = 4 + \sqrt{16 - (x - 16)^2}$$ 5. **Explanation:** - Semi-circles A and B are on the x-axis, so $k=0$. - Semi-circles C and D are shifted up by 4 units, so $k=4$. **Final answers:** $$\boxed{\begin{cases} B: y = \sqrt{16 - (x - 12)^2} \\ C: y = 4 + \sqrt{16 - (x - 8)^2} \\ D: y = 4 + \sqrt{16 - (x - 16)^2} \end{cases}}$$