To determine which statement proves that parallelogram KLMN is a rhombus, we need to recall that a rhombus is a special type of parallelogram where all four sides have equal length. Additionally, the diagonals of a rhombus bisect each other at right angles (perpendicular to each other).
Here are the given details we need 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].
3. The length of [tex]\(\overline{NL}\)[/tex] is [tex]\(\sqrt{8}\)[/tex].
4. The slopes of [tex]\(\overline{LM}\)[/tex] and [tex]\(\overline{KN}\)[/tex] are both [tex]\(\frac{1}{2}\)[/tex].
5. The lengths of [tex]\(\overline{NK}\)[/tex] and [tex]\(\overline{ML}\)[/tex] are both [tex]\(\sqrt{20}\)[/tex].
6. The slope of [tex]\(\overline{KM}\)[/tex] is [tex]\(1\)[/tex].
7. The slope of [tex]\(\overline{NL}\)[/tex] is [tex]\(-1\)[/tex].
Now, let's break down each condition to see what they tell us:
1. Midpoint of diagonals being the same:
This tells us that the diagonals bisect each other, which is a property of parallelograms, not exclusively rhombuses. Hence, this fact alone cannot prove that KLMN is a rhombus.
2. Lengths of diagonals [tex]\(\overline{KM}\)[/tex] and [tex]\(\overline{NL}\)[/tex]:
[tex]\(\overline{KM} = \sqrt{72}\)[/tex] and [tex]\(\overline{NL} = \sqrt{8}\)[/tex]. In a rhombus, though diagonals bisect each other, they need not be equal in length. Therefore, unequal diagonal lengths do not prove or disprove that KLMN is a rhombus.
3. Slopes of the sides [tex]\(\overline{LM}\)[/tex] and [tex]\(\overline{KN}\)[/tex]:
Both have the same slope [tex]\(\frac{1}{2}\)[/tex], indicating they are parallel. But parallelism alone doesn't confirm that all sides are equal in length.
4. Lengths of sides [tex]\(\overline{NK}\)[/tex] and [tex]\(\overline{ML}\)[/tex]:
Both being [tex]\(\sqrt{20}\)[/tex], we know that opposite sides of the parallelogram are equal, which is a property of parallelograms, not exclusively rhombuses.
5. Slopes of diagonals [tex]\(\overline{KM}\)[/tex] and [tex]\(\overline{NL}\)[/tex]:
With [tex]\(\overline{KM}\)[/tex] having a slope of [tex]\(1\)[/tex] and [tex]\(\overline{NL}\)[/tex] a slope of [tex]\(-1\)[/tex], this tells us that these diagonals are perpendicular because their slopes are negative reciprocals of each other.
Therefore, based on the given information, no single condition on its own conclusively demonstrates that KLMN is a rhombus. The necessary condition for a rhombus — all four sides having equal length — is not provided explicitly.
Considering this thorough analysis, we can conclude that:
None of the given statements, on their own, prove that KLMN is a rhombus.
