Rotate 180 Degrees Around The Origin

🔑 Key Concepts

  • A \(180^\circ\) rotation is a half-turn around the origin.
  • The rule for a \(180^\circ\) rotation around the origin is \((x,y)\rightarrow(-x,-y)\).
  • Both coordinates become opposites, but the order of the coordinates does not change.
  • A point and its image are the same distance from the origin, but on opposite sides of the origin.
  • A \(180^\circ\) rotation is a rigid transformation because the figure keeps the same size and shape.

✏️ Worked Examples

🧠 Math Vocabulary

  • Rotation: a transformation that turns a figure around a fixed point.
  • \(180^\circ\) Rotation: a half-turn around the center of rotation.
  • Center of Rotation: the fixed point that a figure rotates around.
  • Origin: the point \((0,0)\), where the \(x\)-axis and \(y\)-axis intersect.
  • Rigid Transformation: a transformation that preserves 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.
  • Opposite Coordinates: coordinates with the same absolute values but opposite signs, such as \((3,2)\) and \((-3,-2)\).
  • Quadrant: one of the four regions of the coordinate plane.

💡 Main Idea

A \(180^\circ\) rotation around the origin sends each point to the opposite side of the origin. The coordinate rule is \((x,y)\rightarrow(-x,-y)\), so both coordinates change signs while staying in the same order.

📚 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

After practicing \(180^\circ\) rotations, students can compare \(90^\circ\), \(180^\circ\), and \(270^\circ\) rotations around the origin and connect rotations to other rigid transformations.

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