1. **State the problem:** Find the scale factor given the corresponding coordinates of points before and after scaling. 2. **Formula:** The scale factor $k$ is found by dividing the coordinates of the image point by the original point: $$k = \frac{x'}{x} = \frac{y'}{y}$$ 3. **First pair of points:** $A(3,5)$ and $A'(6,10)$. Calculate scale factor for $x$-coordinates: $$k_x = \frac{6}{3} = 2$$ Calculate scale factor for $y$-coordinates: $$k_y = \frac{10}{5} = 2$$ Since both are equal, the scale factor is $2$. 4. **Second pair of points:** $A(-1,4)$ and $A'(-3,12)$. Calculate scale factor for $x$-coordinates: $$k_x = \frac{-3}{-1} = 3$$ Calculate scale factor for $y$-coordinates: $$k_y = \frac{12}{4} = 3$$ Since both are equal, the scale factor is $3$. **Final answers:** - For the first pair, scale factor is $2$. - For the second pair, scale factor is $3$.