This figure shows two triangles, A B C and F E D. Side D F measures 6.5 units. Side D E measures 5 units. Segment X Y is drawn parallel to A C from point X on side A B to point Y on side B C. Segment B Y measures 3 units. \[A\]
\[B\]
\[C\]
\[D\]
\[E\]
\[F\]
\[X\]
\[Y\]
\[6.5\]
\[3\]
\[5\]
Note: Figure not drawn to scale.
Triangles 
\[ABC\] and 
\[FED\] shown are congruent. Line segment 
 
\[\overline{XY}\] is parallel to 
 
\[\overline{AC}\], and the length of 
 
\[\overline{BY}=3\]. What is the length of 
 
\[\overline{XY}\]?


We are given two congruent triangles, \( \triangle ABC \) and \( \triangle FED \), and the following information:

- \( DE = 5 \) units
- \( DF = 6.5 \) units
- \( BY = 3 \) units
- \( \overline{XY} \parallel \overline{AC} \)

Since \( \triangle ABC \) and \( \triangle FED \) are congruent, corresponding sides of the triangles are equal. This means:

\[
AC = DF = 6.5 \quad \text{and} \quad AB = FE = 5
\]

Additionally, since \( \overline{XY} \parallel \overline{AC} \), and \( X \) and \( Y \) lie on sides \( AB \) and \( BC \), respectively, \( \triangle XBY \) is similar to \( \triangle ABC \) by the basic proportionality theorem (or Thales' theorem). This implies that the corresponding sides of these triangles are proportional.

The ratio of similarity between the triangles \( XBY \) and \( ABC \) is determined by the proportion:

\[
\frac{BY}{BC} = \frac{XY}{AC}
\]

We know that \( BY = 3 \), and we can find \( BC \) as follows. Since \( \triangle ABC \) and \( \triangle FED \) are congruent, the side \( BC = ED \). From the problem, \( ED = 5 \) units, so \( BC = 5 \) units.

Now, we can substitute into the proportion:

\[
\frac{3}{5} = \frac{XY}{6.5}
\]

Solve for \( XY \):

\[
XY = \frac{3}{5} \times 6.5 = \frac{19.5}{5} = 3.9 \, \text{units}
\]

Thus, the length of \( \overline{XY} \) is \( 3.9 \) units.

---

Do you have any questions or would you like further clarification on any part of the solution?

Here are some related questions to further explore:

1. How would the answer change if \( BY \) were longer than 3 units?
2. Can we find the area of \( \triangle ABC \) given the side lengths?
3. How does the parallel condition between \( XY \) and \( AC \) affect triangle similarity?
4. What are the other corresponding side lengths of the congruent triangles?
5. How can we use congruence postulates to prove the triangles are congruent?

**Tip:** When dealing with similar triangles, always focus on the ratio of corresponding sides for calculating unknown lengths.

This figure shows two triangles, A B C and F E D. Side D F measures 6.5 units. Side D E measures 5 units. Segment X Y is drawn parallel to A C from point X on side A B to point Y on side B C. Segment B Y measures 3 units. Triangles ABC and FED shown are congruent. Line segment XY is parallel to AC, and the length of BY=3. What is the length of XY?

Solve for the length of XY in a problem involving two congruent triangles, ABC and FED, using the basic proportionality theorem and congruence. Includes triangle similarity, side lengths, and proportionality concepts. Suitable for Grades 8-10.

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