To determine which statement proves that the parallelogram KLMN is a rhombus, we need to consider the properties of a rhombus. Here are the key properties relevant to our question:
1. Rhombus Properties:
- All four sides have the same length.
- Diagonals bisect each other at right angles.
Now, let's analyze each given statement to see how they align with these properties:
1. The midpoint of both diagonals is [tex]$(4, 4)$[/tex]:
- This tells us that the diagonals bisect each other, which is a property of any parallelogram, including rectangles, squares, and rhombuses. However, this alone does not confirm that KLMN is a rhombus.
2. The length of [tex]$\overline{K M}$[/tex] is [tex]$\sqrt{72}$[/tex] and the length of [tex]$\overline{N L}$[/tex] is [tex]$\sqrt{8}$[/tex]:
- This gives us the lengths of the diagonals but does not confirm they bisect at right angles or that all sides are equal.
3. The slopes of [tex]$\overline{ LM }$[/tex] and [tex]$\overline{ KN }$[/tex] are both [tex]$\frac{1}{2}$[/tex] and [tex]$NK = ML =$[/tex] [tex]$\sqrt{20}$[/tex]:
- Here, we know the slopes of opposite sides and lengths of two sides, however, the fact that opposite slopes are equal and some side lengths are given does not confirm perpendicularity of diagonals nor equal side lengths across all four sides.
4. The slope of [tex]$\overline{ KM }$[/tex] is 1 and the slope of [tex]$\overline{ NL }$[/tex] is -1:
- Knowing that the slope of [tex]$\overline{ KM }$[/tex] is 1 and the slope of [tex]$\overline{ NL }$[/tex] is -1 tells us that these diagonals are perpendicular because the product of their slopes is -1 (i.e., 1 * -1 = -1). This means [tex]$\overline{KM}$[/tex] is perpendicular to [tex]$\overline{NL}$[/tex], a key property of rhombus diagonals. Perpendicular diagonals confirm that the diagonals bisect each other at right angles, indicating a rhombus when combined with concurrent midpoint.
Based on analyzing these statements, the slope of [tex]$\overline{KM}$[/tex] being 1 and the slope of [tex]$\overline{NL}$[/tex] being -1 proves that the diagonals are perpendicular to each other.
Therefore, the statement: "The slope of [tex]$\overline{KM}$[/tex] is 1 and the slope of [tex]$\overline{NL}$[/tex] is -1." proves that parallelogram KLMN is a rhombus. This is the conclusive evidence we need that KLMN is indeed a rhombus.
