Question

which statement proves that parallelogram klmn is a rhombus?
the mid - point of both diagonals is (4, 4).
the length of $overline{km}$ is $sqrt{72}$ and the length of $overline{nl}$ 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:

Step1: Recall rhombus properties

A parallelogram is a rhombus if its diagonals are perpendicular. The product of the slopes of two - perpendicular lines is - 1.

Step2: Analyze slope - based condition

If the slope of one diagonal is \(m_1\) and the slope of the other diagonal is \(m_2\), for perpendicular lines \(m_1\times m_2=-1\). In the last option, the slope of \(\overline{KM}\) is 1 and the slope of \(\overline{NL}\) is - 1. Since \(1\times(-1)=-1\), the diagonals are perpendicular.

Step3: Analyze other options

• The mid - point of both diagonals being the same only shows that the diagonals bisect each other, which is true for all parallelograms, not just rhombuses.
• The lengths of the diagonals being \(\sqrt{72}\) and \(\sqrt{8}\) does not prove it is a rhombus.
• Equal side - lengths and equal slopes of non - parallel sides only show it is a parallelogram, not necessarily a rhombus.

Answer:

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