Rotate a Triangle 180 Degrees Around the Origin

✏️ Worked Example

Rotate Triangle \(ABC\) \(180^\circ\) Around the Origin

Preimage coordinates:
\(A(-4,1)\), \(B(-4,5)\), \(C(-2,1)\)

Image coordinates:
\(A'(4,-1)\), \(B'(4,-5)\), \(C'(2,-1)\)

The preimage triangle starts in Quadrant II. A \(180^\circ\) rotation around the origin makes a half-turn, so the image lands in the opposite diagonal quadrant: Quadrant IV.

Use the \(180^\circ\) rotation rule:

\((x,y)\rightarrow(-x,-y)\)

Apply the rule to each vertex:

\(A(-4,1)\rightarrow A'(4,-1)\)
\(B(-4,5)\rightarrow B'(4,-5)\)
\(C(-2,1)\rightarrow C'(2,-1)\)

What changes?
The \(x\)- and \(y\)-values keep the same absolute values, but both signs change.

For rotations around the origin, a \(180^\circ\) turn sends points to the opposite diagonal quadrant, not an adjacent quadrant.

🧠 Math Vocabulary

• Rotation: a transformation that turns a figure around a fixed point.
• \(180^\circ\) Rotation: a half-turn around the center of rotation.
• Origin: the point \((0,0)\), where the \(x\)-axis and \(y\)-axis intersect.
• Center of Rotation: the fixed point around which the figure turns.
• Rigid Transformation: a transformation that keeps the same size and shape.
• Image: the new figure after a transformation.
• Preimage: the original figure before a transformation.
• Coordinate Rule: an algebraic rule that shows how ordered pairs change during a transformation.

📚 What You Should Already Know

Students should know how to plot ordered pairs, identify the origin, name the four quadrants, and find opposites of positive and negative numbers.

🚀 What Comes Next

Next, students can compare \(180^\circ\) rotations with \(90^\circ\) clockwise and counterclockwise rotations around the origin.

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