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 \( AB = 5.2 \) cm, \( AC = 6.8 \) cm, and \( BD = 8.2 \) cm, follow these steps:

### Steps to Construct the Parallelogram:

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

2. **Find the Midpoint \( O \) of \( AC \)**:
   - Since the diagonals of a parallelogram bisect each other, locate the midpoint \( O \) of \( AC \). To do this:
     - Use a compass to measure and draw arcs from both endpoints \( A \) and \( C \) such that the arcs intersect above and below the line segment.
     - Connect these intersection points to get the perpendicular bisector, and mark the point \( O \) where it intersects \( AC \). This is the midpoint.

3. **Draw Diagonal \( BD \)**:
   - Measure \( BD = 8.2 \) cm.
   - Since the diagonals bisect each other, the midpoint \( O \) will also be the midpoint of \( BD \).
   - From \( O \), measure \( 4.1 \) cm along the perpendicular bisector on either side to mark points \( B \) and \( D \). \( BO = OD = 4.1 \) cm.

4. **Construct 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, from point \( D \), draw another arc with the same radius \( 5.2 \) cm.
   - The intersection of these arcs with the diagonal line \( BD \) will give you point \( B \) opposite to \( D \).

5. **Draw Lines \( BC \) and \( DA \)**:
   - Connect points \( B \) to \( C \) and \( D \) to \( A \) to complete the parallelogram \( ABCD \).

### Summary:
- \( AB = 5.2 \) cm
- \( AC = 6.8 \) cm (diagonal)
- \( BD = 8.2 \) cm (diagonal)
- The diagonals \( AC \) and \( BD \) bisect each other at \( O \), with \( BO = OD = 4.1 \) cm.

The construction ensures that the opposite sides are equal and parallel, confirming that \( ABCD \) is a parallelogram.

### Would you like to go through any of the steps in more detail or have any additional questions?

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**Related Questions:**

1. How can you verify that the constructed figure is a parallelogram?
2. What are the properties of a parallelogram related to its diagonals?
3. How would the construction change if the diagonals were equal?
4. Can this method be adapted for constructing a rectangle or rhombus?
5. What would be different if \( AB \) and \( CD \) had different lengths?

**Tip:** When constructing shapes, always verify that the required properties hold, like side lengths and angle measures. This helps confirm the accuracy of the construction.

Learn how to construct a unique parallelogram ABCD with sides AB = 5.2 cm, AC = 6.8 cm, and diagonal BD = 8.2 cm. Follow step-by-step instructions and understand the properties of parallelograms and their diagonals. Suitable for students in Grades 7-9.

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