To prove that the parallelogram KLMN is a rhombus, let's analyze the given statements:
1. The midpoint of both diagonals is [tex]$(4,4)$[/tex].
- This condition is typically true for any parallelogram, not just a rhombus. It shows that the diagonals bisect each other at the midpoint.
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].
- Knowing the lengths of the diagonals alone is not sufficient to conclude that a 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].
- This tells us that opposite sides are equal and the slopes are the same, which is true for any parallelogram and does not specifically indicate 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 [tex]$\overline{ KM }$[/tex] and [tex]$\overline{ NL }$[/tex] are perpendicular to each other because the product of their slopes is -1. In a rhombus, the diagonals are always perpendicular bisectors of each other.
Based on the slopes of the diagonals:
- The statement that the slope of [tex]$\overline{ KM }$[/tex] is 1 and the slope of [tex]$\overline{ NL }$[/tex] is -1 proves that KLMN is a rhombus. This is because in a rhombus, the diagonals bisect each other at right angles (i.e., they are perpendicular).
Therefore, the correct statement that proves 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."
