On the coordinate plane, point A (x, -4) is symmetric to point B (4, y) relative to point E (0, -2). Find the value of x and y.

If point A (x, -4) is symmetric to point B (4, y), then point E (0, -2) is the center of the segment | AB |.

The coordinates of the midpoint of the segment are found by the formula:

xE = (xA + xB) / 2;
yE = (yA + yB) / 2.

Substituting the values into the formula, we have:

0 = (x + 4) / 2;
x + 4 = 0;
x = -4.

-2 = (y + (-4)) / 2;
-4 = y – 4;
y = -4 + 4;
y = 0.

Answer: A (-4; -4) and B (4; 0).