Construct Unique quadrilateral
Construct a parallelogram ABCD such that AB = 5.2 cm, AC = 6.8 cm and BD = 8.2 cm Hint: Diagonals of a parallelogram bisect each other.

To construct a unique parallelogram \( ABCD \) with the given dimensions where \( AB = 5.2 \) cm, \( AC = 6.8 \) cm, and \( BD = 8.2 \) cm, you can follow these steps:

### Steps to Construct the Parallelogram:

1. **Draw Diagonal \( AC \)**:
   - Draw a line segment \( AC = 6.8 \) cm. This will be one of the diagonals of the parallelogram.

2. **Construct the Perpendicular Bisector of \( AC \)**:
   - Since the diagonals of a parallelogram bisect each other, the midpoint \( O \) of diagonal \( AC \) will be the point where diagonals \( AC \) and \( BD \) intersect.
   - Find the midpoint \( O \) of \( AC \) by constructing the perpendicular bisector of \( AC \).

3. **Draw Diagonal \( BD \)**:
   - Measure \( BD = 8.2 \) cm.
   - Place the midpoint of \( BD \) at \( O \), and draw a line \( BD \) such that it passes through \( O \). The point \( O \) will bisect \( BD \), so each segment \( BO \) and \( OD \) will be \( 4.1 \) cm.

4. **Construct Points \( B \) and \( D \)**:
   - With \( O \) as the midpoint, mark the points \( B \) and \( D \) on the line \( BD \) such that \( BO = OD = 4.1 \) cm.

5. **Complete the Parallelogram**:
   - Now, use a compass to measure \( AB = 5.2 \) cm.
   - Place the compass at point \( A \) and draw an arc with a radius of \( 5.2 \) cm.
   - Similarly, measure \( CD = 5.2 \) cm with the compass.
   - Place the compass at point \( C \) and draw an arc with a radius of \( 5.2 \) cm to intersect the arc from point \( A \).
   - The intersection of these arcs will give points \( B \) and \( D \), completing the parallelogram \( ABCD \).

### Diagram:
- You should have a parallelogram \( ABCD \) with the specified sides and diagonals.
- \( AB = 5.2 \) cm, \( AC = 6.8 \) cm (diagonal), and \( BD = 8.2 \) cm (diagonal).

### Verification:
- Ensure that the diagonals \( AC \) and \( BD \) bisect each other.
- Verify that opposite sides \( AB \) and \( CD \) as well as \( AD \) and \( BC \) are equal in length.

Would you like a more detailed explanation or assistance with any of the steps?

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**Here are some related questions:**

1. How can you verify the angles of the parallelogram after construction?
2. What properties of a parallelogram are used to ensure the diagonals bisect each other?
3. How would the construction change if \( AB \) was not equal to \( CD \)?
4. Can this method be applied to construct other quadrilaterals like rectangles or rhombuses?
5. What would happen if the diagonals were of the same length?

**Tip:** When constructing geometric shapes, it's always useful to double-check the properties of the shape to ensure accuracy.

Learn how to construct a unique parallelogram ABCD with sides AB = 5.2 cm, AC = 6.8 cm, and diagonals BD = 8.2 cm using geometric construction methods. This problem explores the properties of parallelograms and includes step-by-step instructions for construction.

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