109. Reflection in Origin · flip both signs, stay opposite

Try to break it

Move P to any corner. Notice how P' instantly jumps to the diagonally opposite corner, flipping both its horizontal and vertical signs?

How you build it

Reflect a point in the origin.

• Pick the Point tool. Click exactly where the x-axis and y-axis cross — this is O, the origin (0, 0). We need O as an actual clickable point.
• Click anywhere away from O to drop point P — the point we will reflect through the origin. Try a grid intersection so the coordinates are clean.
• Pick the Line tool. Click P, then click O. A straight line is drawn through both points and extends in both directions — including past O to the opposite side. This is the line P will mirror onto.
• Pick the Arc tool. Click O for the centre, then click P to set the radius. The arc passes through P and curves all the way around — crossing the line you drew at exactly one other point, on the opposite side of O.
• Switch to the Point tool. On the opposite side of O from P, the arc and the line cross at exactly one point. Click that intersection — this is P', the reflection of P through the origin. Because OP' = OP and O lies on PP', O is the midpoint of PP'. If P = (x, y), then P' = (-x, -y) — both coordinates flip sign.

The proof, step by step

Prove that P prime is the reflection of P through the origin.

• Connect P to the origin O and extend the line to P'.
• Measure OP and OP'; they are exactly equal in length.
• Observe that P' sits at (-x, -y), meaning both coordinates flipped signs.
• Therefore, reflection in the origin negates both the x and y coordinates.

Worked example

In a Cartesian plane, if point A has coordinates (3, -4), what are the coordinates of its image A' when reflected in the origin?

Reflection in the origin changes the sign of both coordinates. So, (x, y) becomes (-x, -y). For A(3, -4), A' is (-3, 4).

• (-3, 4) — correct
• (3, 4)
• (-3, -4)
• (4, -3)