Geometric Transformation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Geometric Transformation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A function that maps every point of a geometric figure to a new position, changing its location, orientation, or size.

Moving, rotating, flipping, or stretching a shape to produce a new image of that shape.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Transformations change position or size while potentially preserving shape.

Common stuck point: Rigid transformations (translation, rotation, reflection) preserve size and shape; dilations change size.

Sense of Study hint: Ask yourself: did the shape's size change? If no, it was a rigid transformation. If yes, it was a dilation.

Worked Examples

Example 1

easy
Name and describe the four basic geometric transformations.

Solution

  1. 1
    Step 1: Translation — slides every point by the same vector (a, b): (x,y) \to (x+a, y+b).
  2. 2
    Step 2: Rotation — turns every point around a fixed center by a given angle.
  3. 3
    Step 3: Reflection — flips every point over a line (the axis of reflection).
  4. 4
    Step 4: Dilation — scales every point away from or toward a center by a scale factor k.

Answer

Translation, Rotation, Reflection, Dilation.
Translation, rotation, and reflection are isometries — they preserve distances and angles (rigid motions). Dilation changes size but preserves shape (similarity transformation). Together these transformations form the foundation of geometric transformation theory.

Example 2

medium
Point P(3, -1) is reflected over the y-axis, then translated by (2, 5). Find the final image.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Which geometric transformations are isometries (preserve distances)?

Example 2

hard
Show that a 180° rotation about the origin is equivalent to the transformation (x,y) \to (-x,-y). Verify with the point (3, 4).
← Back to Geometric Transformation Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

shapes