https://file.aiquickdraw.com/mathbot/file/1730211226035i83y2d04.jpg

The problem in the image involves finding the value of \( x \) in a rectangle based on the properties of its diagonals. Here’s the step-by-step breakdown:

1. **Diagonals Bisect Each Other**:
   - Since the diagonals of a rectangle bisect each other, we have the midpoint property:
     \[
     WV = \frac{1}{2} WY
     \]
   - Doubling both sides to find \( WY \):
     \[
     2(WV) = WY
     \]
   - Substituting \( WV = 2x + 1 \) (from the given values):
     \[
     2(2x + 1) = WY
     \]
     \[
     4x + 2 = WY
     \]

2. **Diagonals are Congruent**:
   - In a rectangle, the diagonals are congruent, so:
     \[
     WY = XZ
     \]
   - Substituting values for \( WY \) and \( XZ \) from the previous steps:
     \[
     4x + 2 = 8 - 2x
     \]
   - Solving for \( x \):
     - Bring all \( x \)-terms to one side:
       \[
       4x + 2x = 8 - 2
       \]
       \[
       6x = 6
       \]
     - Divide by 6:
       \[
       x = 1
       \]

Thus, the solution confirms that \( x = 1 \).

Would you like further details or have any questions?

### Related Questions
1. How can we verify the congruency of diagonals in other quadrilaterals?
2. What are the properties of a rectangle that involve its diagonals?
3. Can this method be applied to parallelograms or other quadrilaterals?
4. What happens if \( x \) does not satisfy the equation in such problems?
5. Why do diagonals in a rectangle bisect each other?

**Tip**: For any quadrilateral, understanding diagonal properties is crucial, as they often help identify the shape based on their behavior (e.g., bisecting each other, congruence).

Find the value of x in a rectangle where the diagonals bisect each other and are congruent.

Discover how to find the value of x in a rectangle by using properties of bisecting and congruent diagonals. This geometry problem involves algebraic manipulation and understanding theorems about rectangles. Suitable for Grades 8-10.

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