Question

which statement proves that parallelogram klmn is a rhombus?
the midpoint of both diagonals is (4, 4).
the length of \\( \overline { k m } \\) is \\( \sqrt { 7 2 } \\) and the length of \\( \overline { n l } \\) is \\( \sqrt { 8 } \\).
the slopes of \\( \overline { l m } \\) and \\( \overline { k n } \\) are both \\( \frac { 1 } { 2 } \\) and \\( n k = m l = \sqrt { 2 0 } \\).
the slope of \\( \overline { k m } \\) is 1 and the slope of \\( \overline { n l } \\) is -1.

Explanation:

Step1: Recall the property of a rhombus

A parallelogram is a rhombus if and only if its diagonals are perpendicular.
The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Calculate the slopes of the diagonals

For diagonal \(KM\) with \(K(7,7)\) and \(M(1,1)\):
\(m_{KM}=\frac{7 - 1}{7 - 1}=\frac{6}{6}=1\)
For diagonal \(NL\) with \(N(3,5)\) and \(L(5,3)\):
\(m_{NL}=\frac{3 - 5}{5 - 3}=\frac{-2}{2}=-1\)

Step3: Check the perpendicularity

If two lines with slopes \(m_1\) and \(m_2\) are perpendicular, then \(m_1\times m_2=-1\).
Here, \(m_{KM}\times m_{NL}=1\times(-1)= - 1\)

Answer:

The slope of \(\overline{KM}\) is \(1\) and the slope of \(\overline{NL}\) is \(-1\).