1. **State the problem:** We have two triangles, $\triangle ABC$ and $\triangle EDC$, sharing vertex $C$. Given sides are $AB=9$, $AC=12$, $BC=10$, $ED=k$, $DC=x$, and $DE=27$. We need to find the values of $x$ and $k$. 2. **Identify the relationship:** Since the triangles share vertex $C$ and appear to be similar (implied by the problem), corresponding sides are proportional. The similarity ratio relates sides of $\triangle ABC$ to $\triangle EDC$. 3. **Set up proportions:** Corresponding sides are $AB$ to $ED$, $BC$ to $DC$, and $AC$ to $EC$ (or $DE$ in the second triangle). Given $DE=27$ corresponds to $AC=12$, so the scale factor $r = \frac{27}{12} = \frac{9}{4}$. 4. **Find $x$:** Since $BC=10$ corresponds to $DC=x$, use the scale factor: $$x = 10 \times r = 10 \times \frac{9}{4} = \frac{90}{4} = 22.5$$ 5. **Find $k$:** Since $AB=9$ corresponds to $ED=k$, use the scale factor: $$k = 9 \times r = 9 \times \frac{9}{4} = \frac{81}{4} = 20.25$$ 6. **Final answers:** $$x = 22.5$$ $$k = 20.25$$