1. **Problem statement:** We need to find the coordinates of vertex C' after enlarging triangle ABC with center at (1, 9) and scale factor $\frac{1}{2}$. The original vertex C is at (3, 1). 2. **Formula for enlargement:** If a point $P(x,y)$ is enlarged with center $O(x_0,y_0)$ and scale factor $k$, the image point $P'(x',y')$ is given by: $$x' = x_0 + k(x - x_0)$$ $$y' = y_0 + k(y - y_0)$$ 3. **Apply the formula:** - Center $O = (1,9)$ - Original point $C = (3,1)$ - Scale factor $k = \frac{1}{2}$ Calculate $x'$: $$x' = 1 + \frac{1}{2}(3 - 1) = 1 + \frac{1}{2} \times 2 = 1 + 1 = 2$$ Calculate $y'$: $$y' = 9 + \frac{1}{2}(1 - 9) = 9 + \frac{1}{2} \times (-8) = 9 - 4 = 5$$ 4. **Final answer:** The coordinates of vertex C' are $(2, 5)$.