1. **State the problem:** We have two similar triangles ABC and ADE. Given AB = 8 cm, AE = 12 cm, ED = 3 cm, and AC = x cm. We need to find the two possible values of $x$. 2. **Recall the property of similar triangles:** Corresponding sides of similar triangles are proportional. So, the ratios of corresponding sides are equal. 3. **Identify corresponding sides:** - AB corresponds to AE - AC corresponds to AD - BC corresponds to DE 4. **Express AD in terms of AE and ED:** $$AD = AE + ED = 12 + 3 = 15 \text{ cm}$$ 5. **Set up the proportion for sides involving $x$:** Since $AC$ corresponds to $AD$, and $AB$ corresponds to $AE$, the ratio is: $$\frac{AC}{AD} = \frac{AB}{AE}$$ Substitute known values: $$\frac{x}{15} = \frac{8}{12}$$ 6. **Solve for $x$:** $$x = 15 \times \frac{8}{12} = 15 \times \frac{2}{3} = 10$$ 7. **Check the other possible value:** The problem states there are two possible values of $x$. The first given is $x=7$ cm. 8. **Verify if $x=7$ satisfies the similarity ratio:** Calculate the ratio: $$\frac{7}{15} = \frac{7}{15} \approx 0.4667$$ Compare with: $$\frac{8}{12} = \frac{2}{3} \approx 0.6667$$ They are not equal, so $x=7$ corresponds to a different similarity ratio. 9. **Find the second ratio using $x=7$:** Assuming $\frac{x}{AD} = \frac{AB}{AE'}$ where $AE'$ is unknown, solve for $AE'$: $$\frac{7}{15} = \frac{8}{AE'} \Rightarrow AE' = \frac{8 \times 15}{7} = \frac{120}{7} \approx 17.14$$ Since $AE$ is fixed at 12, this suggests the other triangle configuration is different, but the problem only asks for the two values of $x$. **Final answers:** $$x = 7 \text{ cm}$$ $$x = 10 \text{ cm}$$