1. **Problem Statement:** We have two similar triangles, $\triangle ABC$ and $\triangle BDE$. We know $DE=10$ cm, and the side lengths at vertices are $AB=6$, $BC=8$, and $AC=5$. We need to find the length $DA$ and then find the length $AC$ (which is already given as 5, so we will verify it using similarity). 2. **Similarity and Ratios:** Since $\triangle ABC \sim \triangle BDE$, corresponding sides are proportional. The corresponding sides are: - $AB$ corresponds to $BD$ - $BC$ corresponds to $DE$ - $AC$ corresponds to $BE$ 3. **Using the given lengths:** - $BC = 8$ - $DE = 10$ From similarity, the ratio of sides is constant: $$\frac{BC}{DE} = \frac{8}{10} = 0.8$$ 4. **Find $BD$ using the ratio:** Since $AB$ corresponds to $BD$, $$\frac{AB}{BD} = 0.8 \implies BD = \frac{AB}{0.8} = \frac{6}{0.8} = 7.5$$ 5. **Find $DA$:** Since $D$ lies on $AB$, and $AB=6$, and $BD=7.5$ is longer than $AB$, this suggests a misunderstanding. Actually, $D$ lies on $AB$ between $A$ and $B$, so $BD$ must be less than $AB$. Let's reconsider the ratio: Because $\triangle BDE$ is inside $\triangle ABC$, the smaller triangle's sides are proportional to the larger triangle's sides by a scale factor $k < 1$. Given $DE=10$ and $BC=8$, the scale factor is: $$k = \frac{DE}{BC} = \frac{10}{8} = 1.25$$ This is greater than 1, which contradicts the smaller triangle being inside. 6. **Re-examining the problem:** The problem states $DE=10$ cm, $B=8$, $A=6$, $C=5$. These numbers likely represent side lengths or coordinates. Assuming $AB=8$, $AC=6$, and $BC=5$ (rearranged), then: If $BC=5$ and $DE=10$, then the scale factor is: $$k = \frac{DE}{BC} = \frac{10}{5} = 2$$ This means $\triangle BDE$ is larger than $\triangle ABC$, which contradicts the figure. 7. **Assuming $B=8$, $A=6$, $C=5$ are coordinates or lengths on the graph, and $DE=10$ is given, we can find $DA$ by the segment subtraction:** Since $D$ lies on $AB$, and $AB=6$, then: $$DA = AB - BD$$ But $BD$ corresponds to $BE$ in the smaller triangle, and $BE$ corresponds to $AC$ in the larger triangle. 8. **Final step:** Without exact lengths for $BD$ or $BE$, and given the data, the best we can do is state that the length $DA$ is $6 - BD$ where $BD$ is proportional to $AB$ by the scale factor $k = \frac{DE}{BC} = \frac{10}{8} = 1.25$. Therefore, $$BD = \frac{AB}{k} = \frac{6}{1.25} = 4.8$$ So, $$DA = AB - BD = 6 - 4.8 = 1.2$$ 9. **Length $AC$:** Given as 5 cm, consistent with the problem statement. **Answer:** - $DA = 1.2$ cm - $AC = 5$ cm