Questions: In the figure to the right, triangle ABC and triangle ADE are similar. Find the length of EC.

Solution Steps

Given that triangles $\triangle ABC$ and $\triangle ADE$ are similar, we can use the properties of similar triangles. Similar triangles have corresponding sides in proportion.

Since $\triangle ABC$ and $\triangle ADE$ are similar, the ratio of corresponding sides is equal. We can write the proportion as: $$\frac{AB}{AD} = \frac{BC}{DE} = \frac{AC}{AE}$$

From the diagram:

• $AB = 1$
• $AD = AB + BD = 1 + 9 = 10$
• $BC = 1$
• $DE = 14$

Using the proportion: $$\frac{AB}{AD} = \frac{BC}{DE}$$ Substitute the known values: $$\frac{1}{10} = \frac{1}{DE}$$

Since $DE = 14$, we need to find $EC$. From the similar triangles, we know: $$\frac{BC}{DE} = \frac{1}{14}$$ Since $DE = EC + 9$, we can write: $$\frac{1}{14} = \frac{1}{EC + 9}$$

Cross-multiply to solve for $EC$: $$1 \cdot (EC + 9) = 14 \cdot 1$$ $$EC + 9 = 14$$ $$EC = 14 - 9$$ $$EC = 5$$

Final Answer