To determine which statement proves that parallelogram [tex]\(KLMN\)[/tex] is a rhombus, we need to confirm two things:
1. All sides of the parallelogram are equal.
2. The opposite sides are parallel or have equal slopes.
We are provided with the following information to analyze:
1. The midpoint of both diagonals is [tex]\((4,4)\)[/tex].
2. The length of [tex]\(\overline{KM}\)[/tex] is [tex]\(\sqrt{72}\)[/tex] and the length of [tex]\(\overline{NL}\)[/tex] is [tex]\(\sqrt{8}\)[/tex].
3. The slopes of [tex]\(\overline{LM}\)[/tex] and [tex]\(\overline{KN}\)[/tex] are both [tex]\(\frac{1}{2}\)[/tex], and [tex]\(NK = ML = \sqrt{20}\)[/tex].
4. The slope of [tex]\(\overline{KM}\)[/tex] is 1, and the slope of [tex]\(\overline{NL}\)[/tex] is -1.
Let's analyze these points in detail:
1. The midpoint of both diagonals is [tex]\((4,4)\)[/tex]:
- This statement indicates that the diagonals bisect each other, which is true for all parallelograms, not just rhombuses.
2. The length of [tex]\(\overline{KM}\)[/tex] is [tex]\(\sqrt{72}\)[/tex] and the length of [tex]\(\overline{NL}\)[/tex] is [tex]\(\sqrt{8}\)[/tex]:
- The lengths of the diagonals alone cannot determine whether the parallelogram is a rhombus.
3. The slopes of [tex]\(\overline{LM}\)[/tex] and [tex]\(\overline{KN}\)[/tex] are both [tex]\(\frac{1}{2}\)[/tex] and [tex]\(NK = ML = \sqrt{20}\)[/tex]:
- The fact that the slopes of [tex]\(\overline{LM}\)[/tex] and [tex]\(\overline{KN}\)[/tex] are equal ([tex]\(\frac{1}{2}\)[/tex]) confirms that these sides are parallel.
- Additionally, the sides [tex]\(NK\)[/tex] and [tex]\(ML\)[/tex] being equal ([tex]\(\sqrt{20}\)[/tex]) signifies that the sides opposite each other in the parallelogram are equal, which is a necessary property of a rhombus.
- Finally, all sides being equal (since [tex]\(NK = ML\)[/tex]) truly justifies the shape as a rhombus.
4. The slope of [tex]\(\overline{KM}\)[/tex] is 1 and the slope of [tex]\(\overline{NL}\)[/tex] is -1:
- This implies that the diagonals are perpendicular to each other, which is a property of rhombuses. However, this alone does not directly confirm that all sides are equal.
With the given information, the statement:
"The slopes of [tex]\(\overline{LM}\)[/tex] and [tex]\(\overline{KN}\)[/tex] are both [tex]\(\frac{1}{2}\)[/tex] and [tex]\(NK = ML = \sqrt{20}\)[/tex]"
proves that parallelogram [tex]\(KLMN\)[/tex] is a rhombus since it confirms that the opposite sides are not only parallel but also equal in length, fulfilling the characteristic properties of a rhombus.
