Answer :
To determine whether the parallelogram KLMN is a rhombus, we need to ensure that all four sides are of equal length and the diagonals are perpendicular to each other. Let's analyze the statements given:
1. The midpoint of both diagonals is \((4,4)\).
- This property ensures that the diagonals bisect each other, which is true for all parallelograms but does not specifically prove that KLMN is a rhombus.
2. The length of \(\overline{KM}\) is \(\sqrt{72}\) and the length of \(\overline{NL}\) is \(\sqrt{8}\).
- This statement gives us the lengths of the diagonals, but for a rhombus, we need the lengths of the sides. Additionally, in a rhombus, the diagonals are perpendicular but not necessarily equal.
3. The slopes of \(\overline{ MM }\) and \(\overline{ KN }\) are both \(\frac{1}{2}\) and \(N\)=\(K=\sqrt{20}\).
- This statement seems unclear and does not provide information that can definitively prove that the parallelogram is a rhombus.
4. The slope of \(\overline{KM}\) is 1 and the slope of \(\overline{NL}\) is -1.
- This is the key statement. The slope of \(\overline{KM}\) being \(1\) (which simplifies to \(1\)) and the slope of \(\overline{NL}\) being \(-1\) (which is indeed \(-0.0\)) suggest that these two lines are perpendicular to each other (since the product of their slopes is \(-1\)).
- Since \(\overline{KM}\) and \(\overline{NL}\) represent the diagonals of the parallelogram, their perpendicularity confirms that the parallelogram is a rhombus. This is because in a rhombus, the diagonals are perpendicular bisectors of each other.
Therefore, the correct statement that proves that parallelogram KLMN is a rhombus is:
"The slope of [tex]\(\overline{KM}\)[/tex] is 1 and the slope of [tex]\(\overline{NL}\)[/tex] is -1."
1. The midpoint of both diagonals is \((4,4)\).
- This property ensures that the diagonals bisect each other, which is true for all parallelograms but does not specifically prove that KLMN is a rhombus.
2. The length of \(\overline{KM}\) is \(\sqrt{72}\) and the length of \(\overline{NL}\) is \(\sqrt{8}\).
- This statement gives us the lengths of the diagonals, but for a rhombus, we need the lengths of the sides. Additionally, in a rhombus, the diagonals are perpendicular but not necessarily equal.
3. The slopes of \(\overline{ MM }\) and \(\overline{ KN }\) are both \(\frac{1}{2}\) and \(N\)=\(K=\sqrt{20}\).
- This statement seems unclear and does not provide information that can definitively prove that the parallelogram is a rhombus.
4. The slope of \(\overline{KM}\) is 1 and the slope of \(\overline{NL}\) is -1.
- This is the key statement. The slope of \(\overline{KM}\) being \(1\) (which simplifies to \(1\)) and the slope of \(\overline{NL}\) being \(-1\) (which is indeed \(-0.0\)) suggest that these two lines are perpendicular to each other (since the product of their slopes is \(-1\)).
- Since \(\overline{KM}\) and \(\overline{NL}\) represent the diagonals of the parallelogram, their perpendicularity confirms that the parallelogram is a rhombus. This is because in a rhombus, the diagonals are perpendicular bisectors of each other.
Therefore, the correct statement that proves that parallelogram KLMN is a rhombus is:
"The slope of [tex]\(\overline{KM}\)[/tex] is 1 and the slope of [tex]\(\overline{NL}\)[/tex] is -1."
