Symmetry (including Reflection and Rotation)
187. The 180° Flip · rotating points through the origin
O is always the midpoint of PP'.
180° rotation about origin O: maps P(x, y) to P′(−x, −y). The origin O is the midpoint of segment PP′. This rotation is equivalent to point reflection through O.
What this lesson covers
Try to break it
Drag P anywhere. The line PP' always passes through the origin O, with O splitting PP' into two equal halves. A 180° rotation about O is the same as reflecting through O. Try to drag P so P' lands off that line; you can't.
How you build it
Rotate a point 180° about the origin.
- Pick the Point tool. Click anywhere in the upper-right area to mark point P — the point we will rotate 180° about O.
- Pick the Line tool. Click P, then click O at the origin — the line extends through both, reaching the opposite side where P' will land.
- Pick the Arc tool. Click O for the centre, then click P to set the radius OP. The arc swings around — on the opposite side of O it crosses the line at exactly the right spot for P'.
- Switch to the Point tool. Click where the arc crosses the line on the opposite side of O from P — this is P', the 180° rotation of P. O is the midpoint of PP', and P' has coordinates (−x, −y).
The proof, step by step
Prove that a 180° rotation makes O the midpoint of PP prime.
- A 180° rotation turns the point exactly halfway around the origin.
- This turn flips both the horizontal (x) and vertical (y) directions completely.
- Algebraically, flipping both directions means negating both coordinates: (x, y) → (-x, -y).
- Geometrically, this confirms that O is the midpoint of segment PP', satisfying the 180° rotation property.
Worked example
If point A(3, 4) is rotated 180° about the origin, what are the coordinates of the image A'?
Rotating a point (x, y) by 180° about the origin changes the sign of both coordinates. Thus, A' = (-3, -4).
- (-3, -4) — correct
- (3, -4)
- (-3, 4)
- (4, 3)