1. **Stating the problem:** We have a triangle with side lengths 12.5 cm, 10 cm, and 8.5 cm. A similar triangle has its longest side equal to 20 cm. We need to find the ratio of the areas of the original triangle to the similar triangle. 2. **Formula and rules:** For similar triangles, the ratio of their corresponding sides is the scale factor $k$. The ratio of their areas is the square of the scale factor, i.e., $$\text{Area ratio} = k^2 = \left(\frac{\text{side in original}}{\text{corresponding side in similar}}\right)^2$$ 3. **Find the scale factor:** The longest side of the original triangle is 12.5 cm, and the longest side of the similar triangle is 20 cm. $$k = \frac{12.5}{20} = \frac{12.5}{20}$$ 4. **Simplify the fraction:** $$k = \frac{\cancel{12.5}}{\cancel{20}} = \frac{25/2}{40/2} = \frac{25}{40} = \frac{5}{8} = 0.625$$ 5. **Calculate the area ratio:** $$\text{Area ratio} = k^2 = (0.625)^2 = 0.390625$$ 6. **Conclusion:** The ratio of the areas of the original triangle to the similar triangle is approximately 0.391. **Answer:** b) 0.391