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 1/2 and nk = ml = √20. the length of km is √72 and the length of nl is √8. the slope of km is 1 and the slope of nl is - 1.

Explanation:

To determine which statement proves KLMN is a rhombus, we need to verify all sides are equal. A rhombus has all sides congruent.

Key Calculations:

Vertices: K(7,7), L(5,3), M(1,1), N(3,5).

• KL: Distance = √[(5-7)²+(3-7)²] = √[(-2)²+(-4)²] = √20.
• LM: Distance = √[(1-5)²+(1-3)²] = √[(-4)²+(-2)²] = √20.
• MN: Distance = √[(3-1)²+(5-1)²] = √[(2)²+(4)²] = √20.
• NK: Distance = √[(7-3)²+(7-5)²] = √[(4)²+(2)²] = √20.

All sides equal (√20).

Analysis of Statements:

• "Slopes of LM and KN = 1/2 (opposite sides parallel, parallelogram) and NK = ML = √20 (adjacent sides equal)" confirms it’s a parallelogram with equal sides, hence a rhombus.

Answer:

The slopes of LM and KN are both 1/2 and NK = ML = √20.