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?

---

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.