Geometric Invariance Math Example 1

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Example 1

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

About Geometric Invariance

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

Learn more about Geometric Invariance →

More Geometric Invariance Examples

Example 2 hard

Under a dilation with scale factor [formula] centred at the origin, a circle has centre [formula] an

Example 3 easy

A rectangle is translated 5 units right and 3 units up. Name two properties that are invariant and o

Example 4 hard

The cross-ratio of four collinear points is [formula], an invariant under projective transformations

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