To rotate a point in the coordinate plane counterclockwise about the origin, we can use the following formulas:
x' = x cos θ - y sin θ
y' = x sin θ + y cos θ
where θ is the angle of rotation in radians, and (x, y) are the coordinates of the original point, and (x', y') are the coordinates of the rotated point.
To rotate polygon ABCD 270° counterclockwise, we can apply these formulas to each vertex:
For vertex A(1, 5), we have:
x' = 1 cos (-3π/2) - 5 sin (-3π/2) = 5
y' = 1 sin (-3π/2) + 5 cos (-3π/2) = -1
Therefore, A' is (5, -1).
For vertex B(1, 0), we have:
x' = 1 cos (-3π/2) - 0 sin (-3π/2) = 0
y' = 1 sin (-3π/2) + 0 cos (-3π/2) = -1
Therefore, B' is (0, -1).
For vertex C(-1, -1), we have:
x' = -1 cos (-3π/2) - (-1) sin (-3π/2) = -1
y' = -1 sin (-3π/2) + (-1) cos (-3π/2) = 1
Therefore, C' is (-1, 1).
For vertex D(-4, 2), we have:
x' = -4 cos (-3π/2) - 2 sin (-3π/2) = 2
y' = -4 sin (-3π/2) + 2 cos (-3π/2) = 4
Therefore, D' is (2, 4).
Therefore, the image vertices of polygon ABCD after rotating 270° counterclockwise are A'(5, -1), B'(0, -1), C'(-1, 1), and D'(2, 4).
So the correct answer is A′(5, −1).