To solve the problem, let's use the Basic Proportionality Theorem (or Thales' Theorem). This theorem states that if a line is drawn parallel to one side of a triangle and it intersects the other two sides, then it divides those two sides proportionally.

Here’s the given information:

• $\overline{EF} \parallel \overline{BC}$,
• $BE = 20$,
• $EF = 15$,
• $BC = 27$,
• We are to find $ED$.

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Step 1: Use the proportionality relationship

Since $\overline{EF}$ is parallel to $\overline{BC}$, the segments are proportional: $\frac{BE}{BC} = \frac{EF}{ED}.$

Let $ED = x$.

Substitute the known values: $\frac{20}{27} = \frac{15}{x}.$

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Step 2: Solve for $x$

Cross multiply to solve for $x$: $20x = 27 \cdot 15.$

Simplify: $20x = 405.$

Divide by 20: $x = \frac{405}{20}.$

Simplify: $x = \frac{81}{4}.$

Thus, the length of $ED$ is: $\boxed{\frac{81}{4}} \quad \text{or } 20.25.$

Let me know if you want a more detailed explanation! 😊

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Related Questions:

1. How does the Basic Proportionality Theorem apply to parallel lines in triangles?
2. If the length of $BC$ were different, how would that affect the solution?
3. Why is it important to simplify the radical form in such problems?
4. Can the Basic Proportionality Theorem be used in non-right triangles?
5. What are other applications of proportionality in geometry?

Tip: Always check if the diagram supports the application of proportionality by confirming the parallel condition.