Question: A four-sided figure is resized to create a scaled copy. The lengths of its four sides change as in the table below. Original Figure $8$ $10$ $12$ Scaled Copy $36$ $45$ $54$ Find the constant of proportionality from the original figure to the scaled copy. Express your answer as a fraction in reduced terms.

1. **State the problem:** We are given the side lengths of an original figure and its scaled copy. We need to find the constant of proportionality (scale factor) from the original figure to the scaled copy. 2. **Understand the concept:** The constant of proportionality $k$ relates the original side length $x$ to the scaled side length $y$ by the formula: $$y = kx$$ 3. **Calculate $k$ using the given side lengths:** Using the first pair: $$k = \frac{36}{8}$$ Using the second pair: $$k = \frac{45}{10}$$ Using the third pair: $$k = \frac{54}{12}$$ 4. **Simplify each fraction:** $$\frac{36}{8} = \frac{\cancel{4}9}{\cancel{4}2} = \frac{9}{2}$$ $$\frac{45}{10} = \frac{\cancel{5}9}{\cancel{5}2} = \frac{9}{2}$$ $$\frac{54}{12} = \frac{\cancel{6}9}{\cancel{6}2} = \frac{9}{2}$$ 5. **Conclusion:** All pairs give the same constant of proportionality: $$k = \frac{9}{2}$$ This means the scaled copy is $\frac{9}{2}$ times larger than the original figure in terms of side lengths.