A quadrilateral has vertices at A(−5, 5), B(1, 8), C(4, 2), and D(−2, −2). Use slope to determine if the quadrilateral is a rectangle. Show your work.

Answer 1

To determine if the quadrilateral is a rectangle, we need to check if all four sides are perpendicular to each other.

The slope between two points (x₁, y₁) and (x₂, y₂) can be found using the formula:

m = (y₂ - y₁) / (x₂ - x₁)

Let's calculate the slopes for all four sides AB, BC, CD, and DA:

Slope of AB:
m₁ = (8 - 5) / (1 - (-5)) = 3 / 6 = 1 / 2

Slope of BC:
m₂ = (2 - 8) / (4 - 1) = -6 / 3 = -2

Slope of CD:
m₃ = (-2 - 2) / (-2 - 4) = -4 / (-6) = 2 / 3

Slope of DA:
m₄ = (5 - (-2)) / (-5 - (-2)) = 7 / (-3) = -7 / 3

Now, let's check if any of these slopes are reciprocals of each other (indicating perpendicular lines).

m₁ and m₃ are reciprocals: m₁ * m₃ = (1/2) * (2/3) = 1/3 ≠ -1

Therefore, the quadrilateral is not a rectangle because at least one pair of opposite sides is not perpendicular.