Dilation On The Coordinate Plane Using Slope

🔑 Key Concepts

  • A dilation changes the size of a figure but keeps the same shape.
  • The scale factor, usually written as \(k\), tells how many times farther each image point is from the center of dilation.
  • When the center of dilation is not the origin, each point must be measured from the center of dilation, not from \((0,0)\).
  • Slope steps can help locate image points by repeating the same horizontal and vertical movement from the center of dilation.
  • The dilation rule \( (x,y)\rightarrow \left(a+k(x-a),\,b+k(y-b)\right) \) can be used to calculate image coordinates when the center is \((a,b)\).

✏️ Worked Examples

🧠 Math Vocabulary

  • Dilation: a transformation that enlarges or reduces a figure by a scale factor from a fixed center point.
  • Scale Factor: the number \(k\) that tells how much the figure is enlarged or reduced.
  • Center of Dilation: the fixed point from which all distances are measured during a dilation.
  • Image: the new figure after a transformation. Image points are often labeled with a prime symbol, such as \(A'\), \(B'\), and \(C'\).
  • Preimage: the original figure before the transformation.
  • Non-Rigid Transformation: a transformation that changes the size of a figure. A dilation is non-rigid because side lengths change when \(k\neq1\).
  • Slope Steps: horizontal and vertical movements used to move from one point to another on a coordinate plane.
  • Dilation Formula: the coordinate rule \( (x,y)\rightarrow \left(a+k(x-a),\,b+k(y-b)\right) \), used when the center of dilation is \((a,b)\).

💡 Main Idea

When a figure is dilated from a center that is not the origin, the new points are found by measuring each original point from the center of dilation, multiplying that movement by the scale factor, and then plotting the image point. This can be done visually with slope steps or algebraically using the dilation formula.

📚 What You Should Already Know

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

🚀 What Comes Next

After practicing dilations with centers other than the origin, students can compare dilations, translations, reflections, and rotations as different types of transformations. They can also study similarity, since dilations help explain why similar figures have proportional side lengths.

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