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The problem describes a hotel floor design that combines a rectangle \( DEGF \) and an isosceles triangle \( ABC \). The points \( B \) and \( C \) are midpoints of the sides \( DC \) and \( BE \) of the rectangle, respectively, and asks which outside edges of the floor design are congruent.

### Let's break down the geometry:

- The rectangle \( DEGF \) has opposite sides congruent: 
  - \( DE = FG \) (top and bottom) and
  - \( DF = EG \) (left and right).
  
- The isosceles triangle \( ABC \) implies that:
  - \( AB = AC \), since the triangle is isosceles.

- Since \( B \) and \( C \) are midpoints of \( DC \) and \( BE \), the line segments \( BD \) and \( CE \) are equal.

### Congruent outside edges:
Given that the rectangle's opposite sides are equal, and triangle \( ABC \) is isosceles, the following congruent pairs of outside edges can be identified:
1. \( DF = EG \) (left and right sides of the rectangle),
2. \( DE = FG \) (top and bottom sides of the rectangle),
3. \( AB = AC \) (equal legs of the isosceles triangle).

Thus, the outside edges that are congruent are:
- \( AB = AC \),
- \( DF = EG \),
- \( DE = FG \).

Would you like more details on how these congruencies are derived, or have any further questions?

### Related questions:
1. Why are \( AB \) and \( AC \) equal in an isosceles triangle?
2. How can you prove that \( DF \) is congruent to \( EG \)?
3. What other properties can we deduce about the angles in triangle \( ABC \)?
4. How do midpoints help in defining congruency in geometric figures?
5. What would change if \( B \) and \( C \) were not midpoints?

**Tip**: When dealing with complex geometric shapes, breaking down each figure individually can make it easier to analyze congruent sides and other properties.

The floor design for a new hotel is shown in the figure. The design involves a rectangle, DEGF, joined with an isosceles triangle, ABC. The triangular portion of the floor design is centered where it meets the rectangular portion so that points B and C are the midpoints of DC and BE, respectively. Which outside edges of the floor design are congruent?

In this problem, explore the floor design involving an isosceles triangle joined with a rectangle. Learn how to identify congruent edges by analyzing midpoints and the properties of rectangles and triangles. Suitable for students in Grades 9-10 studying geometry.

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