1. **State the problem:** Verify that triangles \(\triangle ABC\) and \(\triangle DEF\) are similar and find the scale factor from \(\triangle ABC\) to \(\triangle DEF\). 2. **Given side lengths:** - \(\triangle ABC\): \(AB = 15\), \(BC = 18\), \(AC = 12\) - \(\triangle DEF\): \(DE = 10\), \(EF = 12\), \(DF = 8\) 3. **Check similarity:** Triangles are similar if their corresponding sides are proportional. 4. **Calculate ratios of corresponding sides:** - \(\frac{AB}{DE} = \frac{15}{10} = \frac{3}{2} = 1.5\) - \(\frac{BC}{EF} = \frac{18}{12} = \frac{3}{2} = 1.5\) - \(\frac{AC}{DF} = \frac{12}{8} = \frac{3}{2} = 1.5\) 5. Since all three ratios are equal, \(\triangle ABC \sim \triangle DEF\). 6. **Find the scale factor \(k\) from \(\triangle ABC\) to \(\triangle DEF\):** \[ k = \frac{\text{side in } \triangle DEF}{\text{corresponding side in } \triangle ABC} = \frac{10}{15} = \frac{2}{3} \approx 0.6667 \] 7. **Final answer:** The scale factor from \(\triangle ABC\) to \(\triangle DEF\) is \(k = \frac{2}{3}\).