We are tasked with finding the length of $AE$ in the given diagram, where $BC \parallel DE$, and the dimensions of certain segments are labeled.

Steps to Solve:

1. Understand the setup: Since $BC \parallel DE$, the triangles $\triangle ABC$ and $\triangle ADE$ are similar by the AA similarity criterion. This means the ratios of their corresponding sides are equal.

2. Set up the ratio of corresponding sides:

   • From the diagram:

     • $AB = 6 \, \text{in.}$, $AD = 1 \, \text{in.}$, and $AC = AE + EC$.
     • $BC = 3 \, \text{in.}$.

   • By similarity: $\frac{AB}{AD} = \frac{BC}{DE} = \frac{AC}{AE}.$

3. Determine the values:

   • Using the given segment lengths: $\frac{AB}{AD} = \frac{6}{1} = 6.$

   • Let $AE = x$. Then, $EC = 3 \, \text{in.}$, so $AC = x + 3$.

4. Apply the similarity ratio:

   • From the ratio of the sides: $\frac{AC}{AE} = \frac{AB}{AD}.$ Substituting: $\frac{x + 3}{x} = 6.$

5. Solve for $x$: $x + 3 = 6x.$ $3 = 5x.$ $x = \frac{3}{5}.$

Thus, $AE = \frac{3}{5} \, \text{in.}$.

Final Answer:

$AE = \frac{3}{5} \, \text{in.} \, (0.6 \, \text{in.}).$

Would you like a step-by-step explanation of any part?

Related Questions:

1. How can you prove that $\triangle ABC$ and $\triangle ADE$ are similar?
2. Why does the parallel condition $BC \parallel DE$ guarantee similarity between the triangles?
3. How do we calculate the length of $AC$ once $AE$ is known?
4. What would happen to $AE$ if the length of $BC$ was increased to 5 inches?
5. Can this method be extended to 3D shapes with parallel planes?

Tip:

For problems involving parallel lines and triangles, always check for similarity to set up proportional relationships.