Geometric Invariance Examples in Math

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

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 property or measurement of a geometric figure that remains unchanged when a particular transformation is applied.

What stays exactly the same when you move, rotate, or flip a shape? Those unchanging things are invariants.

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: Invariants help classify transformations and identify essential properties.

Common stuck point: Different transformations preserve different things: rigid motions preserve distance; dilations preserve angles but not lengths.

Sense of Study hint: Make a checklist: does the transformation preserve distances? Angles? Area? Check each one to identify what stays the same.

Worked Examples

Example 1

medium
A triangle is reflected across the y-axis and then rotated 90° counterclockwise about the origin. Which properties are invariant: (a) side lengths, (b) angle measures, (c) vertex orientation (CW vs CCW), (d) x-coordinates of vertices?

Solution

  1. 1
    Step 1: Both reflection and rotation are isometries (distance-preserving), so side lengths are invariant. ✓
  2. 2
    Step 2: Isometries preserve angle measures. ✓
  3. 3
    Step 3: Reflection reverses orientation; rotation preserves it. The composition of one reflection and one rotation reverses orientation overall. NOT invariant. ✗
  4. 4
    Step 4: The x-coordinates of vertices change under both operations in general. NOT invariant. ✗

Answer

Invariant: (a) side lengths and (b) angle measures. Not invariant: (c) orientation, (d) x-coordinates.
Geometric invariance identifies what stays the same under a transformation. Isometries preserve distances and angles, but a reflection reverses orientation. Coordinates are position-dependent and change unless the transformation happens to fix them.

Example 2

hard
Under a dilation with scale factor k \neq 1 centred at the origin, a circle has centre (a, b) and radius r. Identify which properties of the circle are invariant and which are not.

Practice Problems

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

Example 1

easy
A rectangle is translated 5 units right and 3 units up. Name two properties that are invariant and one that is NOT invariant under this translation.

Example 2

hard
The cross-ratio of four collinear points is (AC \cdot BD)/(BC \cdot AD), an invariant under projective transformations. If A, B, C, D are at positions 0, 1, 3, 6 on a number line, compute the cross-ratio.
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Background Knowledge

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

transformation geo