To determine which statement proves that parallelogram KLMN is a rhombus, let us go through the given options one by one:
1. The midpoint of both diagonals is \((4,4)\): While the midpoint of the diagonals being the same indicates that the diagonals bisect each other, this property is true for any parallelogram, not just a rhombus. Hence, this statement alone does not prove that KLMN is a rhombus.
2. The length of \(\overline{KM}\) is \(\sqrt{72}\) and the length of \(\overline{NL}\) is \(\sqrt{8}\): For a parallelogram to be a rhombus, all sides must be of equal length. If KM and NL have different lengths, then KLMN cannot be a rhombus. Thus, this statement proves that KLMN is not a rhombus.
3. The slopes of \(\overline{LM}\) and \(\overline{KN}\) are both \(\frac{1}{2}\) and \(NK = ML = \sqrt{20}\): This statement indicates two important things:
- The slopes of \(\overline{LM}\) and \(\overline{KN}\) being equal means these lines are parallel.
- \(NK\) and \(ML\) being equal in length indicates that at least these two sides are of equal length.
Since it's given that the slopes are the same and the lengths of opposing sides are equal, it suggests that:
- The sides of the parallelogram are equal in length and opposite sides are parallel.
- Under these conditions, \(KLMN\) has equal side lengths, which is a defining property of a rhombus.
4. The slope of \(\overline{KM}\) is 1 and the slope of \(\overline{NL}\) is -1: This indicates that the diagonals are perpendicular to each other (since the product of their slopes is -1). However, while perpendicular diagonals are a characteristic of a rhombus, it's not sufficient alone without knowing the sides are equal. This means that this statement, while suggestive, does not conclusively prove that KLMN is a rhombus.
Therefore, considering the properties and conditions of all sides being equal and parallel, the correct statement that proves the parallelogram KLMN is a rhombus is:
3. The slopes of \(\overline{LM}\) and \(\overline{KN}\) are both \(\frac{1}{2}\) and \(NK = ML = \sqrt{20}\).
Hence, the correct option that proves parallelogram KLMN is a rhombus is (C).
