Write a coordinate proof. Prove quadrilateral KLMN, having vertices K(–5, –4), L(0, 8), M(7, 4), and N(8, –4) is not a parallelogram.

From the calculation,

KL = 13

NM = √65

Hence KL ≠ NM

Therefore, the quadrilateral KLMN is not a Parallelogram.

Step-by-step explanation:

K(–5, –4), L(0, 8), M(7, 4), and N(8, –4)

Where we have vertices, (x1, y1) , (x2, y2)

We use the formula

√(x2 - x1)² + (y2 - y1)²

Side KL = K(–5, –4), L(0, 8)

√(x2 - x1)² + (y2 - y1)²

√(0 - (-5))² + (8 - (-4))²

√ 5² + 12²

√25 + 144

= √169

= 13

Side KN = K(–5, –4), N(8, –4)

√(x2 - x1)² + (y2 - y1)²

√(8 -(-5))² + (-4 - (-4))²

√13² + 0²

√169

=13

Side MN = M(7, 4), N(8, –4)

√(x2 - x1)² + (y2 - y1)²

√(8 -7)² + (-4 - 4)²

√1² + (-8)²

√1 + 64

√65

Side LM = L(0, 8), M(7, 4),

√(x2 - x1)² + (y2 - y1)²

√(7-0)² + (4 - 8)²

√(49)² +(-4)²

√ 49 + 16

√65

We were asked in the above question to prove that the quadrilateral with the given vertices is not a Parallelogram.

One of the characteristics of a Parallelogram is that the opposite sides are parallel and congruent to one another. This means that, the opposite sides are similar .

For the Quadrilateral KLMN above to be a Parallelogram, this means

KL = NM

From the above calculation,

KL = 13

NM = √65

Hence KL ≠ NM

Therefore, the quadrilateral KLMN is not a Parallelogram.