1. **Problem Statement:** We are given three triangles: $\triangle ABC$, $\triangle DEF$, and $\triangle IGH$ with side lengths: - $\triangle ABC$: $AC=6$, $CB=4$, $AB=7$ - $\triangle DEF$: $DE=12$, $EF=7$, $DF=9$ - $\triangle IGH$: $IH=9$, $HG=6$, $IG=10.5$ We need to analyze $\triangle DEF$ only. 2. **Goal:** Determine if $\triangle DEF$ is similar to $\triangle ABC$ by comparing their side lengths. 3. **Formula and Rules:** Triangles are similar if their corresponding sides are proportional, i.e., the ratios of corresponding sides are equal: $$\frac{DE}{AC} = \frac{EF}{CB} = \frac{DF}{AB}$$ 4. **Calculate ratios:** $$\frac{DE}{AC} = \frac{12}{6} = 2$$ $$\frac{EF}{CB} = \frac{7}{4} = 1.75$$ $$\frac{DF}{AB} = \frac{9}{7} \approx 1.2857$$ 5. **Interpretation:** Since $2 \neq 1.75 \neq 1.2857$, the ratios are not equal. 6. **Conclusion:** $\triangle DEF$ is **not similar** to $\triangle ABC$ because the side lengths are not proportional. **Final answer:** $\triangle DEF$ is not similar to $\triangle ABC$.