If the line segment XY with endpoints X(3, 1) and Y(2,-2) is rotated 90° counterclockwise about (-6, 4), the endpoints of X'Y' are X' (-3, 13) and Y' (0, 12).
In Euclidean Geometry, the rotation of a point 90° about a point in a counterclockwise (anticlockwise) direction would produce a point that has the following coordinates;
(x, y) → (-(y - b) + a, (x - a) + b)
By applying a rotation of 90° counterclockwise about the point (-6, 4) ≡ (a, b) to the vertices of triangle ABC, the coordinates of the vertices of its image are as follows:
(x, y) → (-(y - b) + a, (x - a) + b)
X (3, 1) → (-(1 - 4) - 6, (3 + 6) + 4) = (3 - 6, 9 + 4)) = X' (-3, 13).
Y (2, -2) → (-(-2 - 4) - 6, (2 + 6) + 4) = (6 - 6, 8 + 4)) = Y' (0, 12).
In conclusion, we can logically deduce that the endpoints of X'Y' are X' (-3, 13) and X' (0, 12).