Rotate a Point About the Origin

Rotate a Point About the Origin

When a point is rotated about the origin, its distance from the origin stays the same, but its coordinates change position according to the direction and angle of rotation.

1. Identify the starting point

Write the point as ((x, y)). Keep track of the signs carefully, since rotation often changes them.

2. Use the rotation rule

For common rotations about the origin:

• 90° counterclockwise: ((x, y) \to (-y, x))
• 90° clockwise: ((x, y) \to (y, -x))
• 180°: ((x, y) \to (-x, -y))

If the problem states a different direction, interpret it carefully before applying the rule.

3. Substitute and simplify

Replace (x) and (y) with the coordinates of the original point, then simplify the signs.

4. Check your answer

A quick check is whether the new point is the correct quarter-turn or half-turn from the original point and whether the distance from the origin is unchanged.

Example check

If ((3, 4)) is rotated 90° counterclockwise, the result is ((-4, 3)). The distance from the origin is still the same.

Always give the final answer as an ordered pair in simplest form.