- Recall that a parallelogram is a rhombus if and only if its diagonals are perpendicular.
- Evaluate the conditions: Statement 1 (lengths) and Statement 2 (midpoint) do not prove the diagonals are perpendicular.
- Calculate the product of the slopes of the diagonals from Statement 3: \(1 \cdot (-1) = -1\).
- Conclude that Statement 3 is sufficient to prove the parallelogram is a rhombus.
- The final answer is \(\boxed{3}\).
Explanation
- Analyze the definition of a rhombus and the given conditions.
A parallelogram \(KLMN\) is defined by its four vertices. A well-known geometric property is that any parallelogram is a rhombus if and only if its diagonals are perpendicular to each other.
Let the diagonals be \(\overline{KM}\) and \(\overline{NL}\). We are asked to determine which of the three given conditions is sufficient to prove that the parallelogram \(KLMN\) is a rhombus:
- The length of \(\overline{KM}\) is \(\sqrt{72}\) and the length of \(\overline{NL}\) is \(\sqrt{8}\).
- The midpoint of both diagonals is \((4, 4)\).
- The slope of \(\overline{KM}\) is \(1\) and the slope of \(\overline{NL}\) is \(-1\).
- Evaluate Statement 1: Diagonals lengths.
Statement 1 provides the lengths of the diagonals \(\overline{KM}\) and \(\overline{NL}\). The lengths of the diagonals in a parallelogram do not dictate the angle at which they intersect. A parallelogram can have diagonals of any lengths \(\sqrt{72}\) and \(\sqrt{8}\) without being a rhombus. Therefore, this condition is insufficient.
- Evaluate Statement 2: Diagonals midpoint.
Statement 2 states that the midpoint of both diagonals is \((4, 4)\). A fundamental property of every parallelogram is that its diagonals bisect each other, meaning they always share the same midpoint. Thus, this statement is true for any parallelogram and does not provide any specific information to identify it as a rhombus. Therefore, this condition is insufficient.
- Evaluate Statement 3: Diagonals slopes.
Statement 3 provides the slopes of the diagonals: \(m_{KM} = 1\) and \(m_{NL} = -1\). For two lines to be perpendicular, the product of their slopes must be \(-1\). Calculating the product of the slopes of the diagonals:
\[m_{KM} \cdot m_{NL} = 1 \cdot (-1) = -1\]
Since the product of the slopes is \(-1\), the diagonals \(\overline{KM}\) and \(\overline{NL}\) are perpendicular. As a parallelogram with perpendicular diagonals is a rhombus, this statement is sufficient to prove that \(KLMN\) is a rhombus.
- Conclusion.
Based on the evaluation, Statement 3 is the only one that directly demonstrates that the diagonals of the parallelogram are perpendicular, which is a sufficient condition to identify the parallelogram as a rhombus.
The final answer is \(\boxed{3}\).
Examples
Understanding the properties of parallelograms and rhombi is useful in fields like computer graphics and structural engineering. For example, if you are designing a grid pattern for a floor tile or a truss system for a bridge, knowing that a parallelogram with perpendicular diagonals forms a rhombus allows you to ensure symmetry and balance in your construction. By checking the slopes of the structural beams (diagonals), you can quickly verify if your design forms a rhombus, ensuring the stability and aesthetic appeal of your project.