Question

which statement proves that parallelogram klmn is a rhombus? the mid - point of both diagonals is (4, 4). the slopes of lm and kn are both 1/3 and nk = ml=sqrt{20}. the slope of km is 1 and the slope of nl is - 1.

Explanation:

Step1: Recall rhombus properties

A parallelogram is a rhombus if its diagonals are perpendicular. The slope of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. If the product of the slopes of two lines is - 1, the lines are perpendicular.

Step2: Analyze each option

• Option 1: The mid - point of both diagonals being the same only shows that the diagonals bisect each other, which is a property of all parallelograms, not just rhombuses.
• Option 2: Equal side lengths and equal slopes of opposite sides show it is a parallelogram, but not necessarily a rhombus.
• Option 3: If the slope of $KM$ is $m_1 = 1$ and the slope of $LN$ is $m_2=-1$, then $m_1\times m_2=1\times(- 1)=-1$. This means the diagonals $KM$ and $LN$ are perpendicular. A parallelogram with perpendicular diagonals is a rhombus.

Answer:

The slope of KM is 1 and the slope of LN is - 1.