Question

which statement proves that parallelogram klmn is a rhombus?

the midpoint of both diagonals is (4, 4).
the length of \\(km\\) is \\(\sqrt{72}\\) and the length of \\(nl\\) is \\(\sqrt{8}\\).
the slopes of \\(\overline{lm}\\) and \\(\overline{kn}\\) are both \\(\frac{1}{2}\\) and \\(nk = ml = \sqrt{20}\\).
the slope of \\(\overline{km}\\) is 1 and the slope of \\(\overline{nl}\\) is -1.

Explanation:

Identify the properties of a rhombus and its diagonals

$$ \text{A parallelogram is a rhombus if and only if its diagonals are perpendicular.} $$

Calculate the slopes of the diagonals KM and NL

$$
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Determine the perpendicular relationship

$$ \text{Slope of } \overline{KM} \times \text{Slope of } \overline{NL} = 1 \times (-1) = -1 \implies \overline{KM} \perp \overline{NL} $$

Answer:

• The midpoint of both diagonals is (4, 4).
• The length of KM is \(\sqrt{72}\) and the length of NL is \(\sqrt{8}\).
• The slopes of \(\overline{LM}\) and \(\overline{KN}\) are both \(\frac{1}{2}\) and \(NK = ML = \sqrt{20}\).
• The slope of \(\overline{KM}\) is 1 and the slope of \(\overline{NL}\) is \(-1\). (Correct answer)