Lesson 11.2: Rotations & Dilations Applied Problems

Learning Objectives

• Apply rotations around the origin and other points.
• Solve problems involving dilations and scale factors.
• Combine transformations to solve multi-step geometry problems.

Topics Covered

• Rotation rules in coordinate geometry
• Dilation formulas and scale factor interpretation
• Composite transformations
• Real-world transformation modeling

Key Terms

• Rotation: A transformation that turns a figure around a fixed point.
• Dilation: A transformation that resizes a figure using a scale factor.
• Scale Factor: The ratio used to enlarge or reduce a figure.
• Composite Transformation: Two or more transformations performed sequentially.

Lesson Examples

Example 1: Rotation About the Origin

1. Rotate point \( (3, 2) \) 90° counterclockwise.
2. Rule: \( (x,y) \rightarrow (-y,x) \)
3. New point: \( (-2,3) \)

Example 2: Dilation

1. Dilate triangle with vertex \( (4,2) \) using scale factor 2.
2. Multiply coordinates by 2.
3. New point: \( (8,4) \)

Example 3: Composite Transformation

1. Rotate point \( (2,1) \) 180° about origin.
2. Apply dilation scale factor 3.
3. Final point: \( (-6,-3) \)

Practice Problems

1. Rotate point \( (5,1) \) 90° clockwise.
2. Dilate triangle with vertices \( (1,2), (3,2), (2,5) \) using scale factor 0.5.
3. Perform rotation 180° followed by dilation with factor 4 on point \( (1,3) \).

Tips & Common Mistakes

• Memorize rotation coordinate rules.
• Apply scale factor to every coordinate during dilation.
• Follow transformation order carefully in composite problems.

Summary

Rotations change orientation while dilations change size. Combining transformations allows complex geometric modeling and real-world problem solving.

Challenge Problems

1. A square with side length 4 is dilated by 1.5 and rotated 90°. Find new vertex positions.
2. Design a composite transformation that maps \( (2,2) \) to \( (-8,8) \).
3. Explain how dilations affect perimeter and area measurements.