To construct a rhombus $PQRS$ with each side measuring $4.5 \, \text{cm}$ and one diagonal measuring $6 \, \text{cm}$, you can follow these steps:

Step-by-step Construction

1. Draw Diagonal PR:

   • Draw a line segment $PR$ of length $6 \, \text{cm}$. This will be one of the diagonals of the rhombus.

2. Find the Perpendicular Bisector of PR:

   • Since the diagonals of a rhombus bisect each other at right angles, find the midpoint $O$ of $PR$.
   • Draw a perpendicular bisector of $PR$ through the midpoint $O$.

3. Determine Length of Diagonal SQ:

   • Let the other diagonal be $SQ$, and since it bisects $PR$, it creates two right triangles $POS$ and $QOR$.
   • Using the Pythagorean theorem in one of the right triangles, we have: $\left(\frac{SQ}{2}\right)^2 + 3^2 = 4.5^2$ Solving for $SQ$, we get: $\left(\frac{SQ}{2}\right)^2 = 4.5^2 - 3^2$ $\frac{SQ}{2} = \sqrt{4.5^2 - 3^2}$ $SQ = 2 \times \sqrt{4.5^2 - 3^2} \approx 5.66 \, \text{cm}$

4. Draw Diagonal SQ:

   • Measure $SQ \approx 5.66 \, \text{cm}$ and mark points $S$ and $Q$ on the perpendicular bisector such that $O$ is the midpoint of $SQ$.

5. Complete the Rhombus:

   • Connect $S$ to $P$ and $R$, and $Q$ to $P$ and $R$ to form the rhombus $PQRS$.

Diagram

Here is the diagram illustrating the construction:

plaintextCopy code
     S
    / \
   /   \
  /     \
 P-------R
  \     /
   \   /
    \ /
     Q

In this diagram:

• $PR = 6 \, \text{cm}$ is the given diagonal.
• $PQ = QR = RS = SP = 4.5 \, \text{cm}$ are the sides of the rhombus.
• $SQ \approx 5.66 \, \text{cm}$ is the other diagonal.

This method uses the property that the diagonals of a rhombus bisect each other at right angles and divides each diagonal into equal halves.

If you have any questions or would like more details on any part of the process, feel free to ask!

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Further Exploration

1. What are the properties of the diagonals of a rhombus?
2. How can the area of a rhombus be calculated using its diagonals?
3. Can you construct the rhombus using only a compass and straightedge?
4. How do the angles in a rhombus relate to its diagonals?
5. What happens to the shape of a rhombus if one diagonal is longer than the sum of its sides?
6. Can you identify the type of quadrilateral formed if one diagonal is zero?
7. How does a rhombus differ from a square, and under what conditions does a rhombus become a square?
8. Can a rhombus have all angles equal to each other? If so, what special type of rhombus is this?

Tip: When constructing geometrical figures, understanding the properties of the shape can simplify the process and ensure accuracy.