Lecture Notes: Dilations and Transformations

Introduction to Dilations

• Focus on dilations in geometric transformations.
• Comparison of triangles A, B, and C, noticed to have the same angles but different sizes.

Understanding Dilations

• Definition: A dilation is a non-rigid transformation where each point moves along a line and changes its distance from a fixed point, known as the center of dilation.
• Characteristics:
  • Maintains angles but changes the size of the shape.
  • All distances are multiplied by the same scale factor.

Transformation Examples

• Triangle B transforms to Triangle A and C through dilation.
• Circular grids can assist in visualizing dilations due to consistent distance increment.
  • Example with radius increment on circular grids.

Performing Dilations

• Requirements:
  1. Center of dilation
  2. Scale factor
  3. Point on the original figure
• Example:
  • Center P, scale factor of 2.
  • PA' = 6 units (double the original PA of 3 units).

Dilation without a Grid

• Center point A with points B, C, and D.
• Point C is the dilation of B with a scale factor of 2 (distance from A to C is twice the distance from A to B).
• For scale factors < 1, dilation brings points closer to the center (Example: Point D is one-third the distance of A to B).

Using Grids for Dilations

• Square Grids: Useful when center and points lie on grid points to easily measure distances by counting grid units.
• Example with scale factor 3/2:
  • Point Q (4 left, 2 down from P).
  • Q' (6 left, 3 down from P) calculated by multiplying distances from P.

Coordinate Grids and Dilations

• Steps:
  1. Identify coordinates of original triangle vertices.
  2. Apply scale factor to each coordinate.
  3. Draw segments for the new, dilated triangle.
• Example with Center (0, 0) and Scale Factor 2:
  • Original Coordinates: (-1, -2), (3, 1), (2, -1).
  • New Coordinates: (-2, -4), (6, 2), (4, -2).

Conclusion

• First video in the series covering dilations, similarity, and introduction to slopes.