Dilation - When The Center Of Dilation Is NOT The Origin

🔑 Key Concepts

  • When the center of dilation is not the origin, every point must be measured from the center of dilation.
  • A scale factor greater than \(1\) creates an enlargement.
  • Each image point lies on the same ray from the center of dilation as its matching preimage point.
  • After one image point is found, corresponding side lengths can help complete the new similar figure.
  • The image and preimage are similar figures, so corresponding sides are multiplied by the same scale factor.

✏️ Worked Examples

🧠 Math Vocabulary

  • Dilation: a transformation that enlarges or reduces a figure from a fixed center point.
  • Center of Dilation: the fixed point used as the starting point for measuring distances during a dilation.
  • Scale Factor: the value \(k\) that tells how many times larger or smaller the image will be compared to the preimage.
  • Enlargement: a dilation with a scale factor greater than \(1\).
  • Preimage: the original figure before a transformation.
  • Image: the new figure after a transformation, often labeled with prime symbols such as \(A'\), \(B'\), and \(C'\).
  • Corresponding Sides: matching sides of two figures that are in the same relative position.
  • Similar Figures: figures with the same shape and proportional side lengths.
  • Non-Rigid Transformation: a transformation that changes the size of a figure. A dilation is non-rigid when \(k\neq1\).

💡 Main Idea

When dilating a figure from a center that is not the origin, measure each point from the center of dilation. You can repeat the movement from the center, use the dilation formula, or use corresponding side lengths once part of the image has been located.

📚 What You Should Already Know

Students should know how to plot ordered pairs, count horizontal and vertical movement on a coordinate plane, multiply integers, identify corresponding sides, and understand that a scale factor greater than \(1\) creates an enlargement.

🚀 What Comes Next

After learning dilations with centers other than the origin, students can compare transformations, identify scale factors from graphs, and connect dilations to similarity and proportional side lengths.

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