Question

I will prove that quadrilateral KLMN is a parallelogram by demonstrating that _ The vertices are K (−3, −1), L (−2, −8), M (1, −2), and N (0, 5).

Answer 1

You can show KLMN is a parallelogram by proving each pair of opposite sides are parallel (equal slopes).

Compute slopes:

- KL: K(−3,−1) to L(−2,−8)
slope = (−8 − (−1)) / (−2 − (−3)) = (−7) / 1 = −7

- MN: M(1,−2) to N(0,5)
slope = (5 − (−2)) / (0 − 1) = 7 / (−1) = −7

So KL ∥ MN.

- LM: L(−2,−8) to M(1,−2)
slope = (−2 − (−8)) / (1 − (−2)) = 6 / 3 = 2

- KN: K(−3,−1) to N(0,5)
slope = (5 − (−1)) / (0 − (−3)) = 6 / 3 = 2

So LM ∥ KN.

Since both pairs of opposite sides are parallel, KLMN is a parallelogram.

(As an alternative check, the diagonals KM and LN have the same midpoint: midpoint KM = ((−3+1)/2, (−1+(−2))/2) = (−1, −3/2), midpoint LN = ((−2+0)/2, (−8+5)/2) = (−1, −3/2). Diagonals bisect each other, which also implies a parallelogram.)