To find the length of segment $EC$, let’s examine the problem based on the similarity of triangles.

Given Information

1. Triangles $\triangle ABC$ and $\triangle ADE$ are similar.
2. We are given:
   • $AB = 1$
   • $DE = 5$
   • $AE = 6$

Solution

Since $\triangle ABC \sim \triangle ADE$, the corresponding sides of these triangles are proportional. This implies: $\frac{AB}{DE} = \frac{BC}{EC} = \frac{AC}{AE}$

1. Step 1: Set up the proportion using $\frac{AB}{DE} = \frac{AC}{AE}$: $\frac{1}{5} = \frac{AC}{6}$

2. Step 2: Solve for $AC$: $AC = \frac{1 \cdot 6}{5} = \frac{6}{5}$

3. Step 3: Use the similarity ratio to find $EC$: Since $\frac{BC}{EC} = \frac{AB}{DE} = \frac{1}{5}$, we can set up another proportion: $\frac{1}{5} = \frac{BC}{EC}$

Given $BC = 1$, we substitute: $\frac{1}{5} = \frac{1}{EC}$

4. Step 4: Solve for $EC$: $EC = 5$

Answer

The length of $EC$ is $\boxed{5}$.

Would you like more details on any step?

Follow-Up Questions:

1. How can we determine if two triangles are similar based on their angles and side ratios?
2. What are other properties that become proportional when triangles are similar?
3. How would the approach change if only part of the triangle lengths were given?
4. Could we use trigonometry to verify the lengths in this triangle?
5. How does the similarity of triangles apply in real-life situations?

Tip:

In similar triangles, always set up the proportion carefully between corresponding sides, as small errors in matching sides can affect the final answer.