Invariants Under Transformation Math Example 1

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Example 1

easy
A triangle with vertices at (1,1)(1,1), (4,1)(4,1), and (1,5)(1,5) is translated by the vector ⟨3,−2⟩\langle 3, -2 \rangle. Which properties are invariant under this translation?

Solution

  1. 1
    Apply the translation to each vertex: (1,1)→(4,−1)(1,1) \to (4,-1), (4,1)→(7,−1)(4,1) \to (7,-1), (1,5)→(4,3)(1,5) \to (4,3).
  2. 2
    Compute side lengths before: 9+0=3\sqrt{9+0}=3, 0+16=4\sqrt{0+16}=4, 9+16=5\sqrt{9+16}=5. After: 9+0=3\sqrt{9+0}=3, 0+16=4\sqrt{0+16}=4, 9+16=5\sqrt{9+16}=5. Side lengths are preserved.
  3. 3
    Angles depend only on side lengths (by the Law of Cosines), so angles are also preserved.
  4. 4
    Area =12(3)(4)=6= \frac{1}{2}(3)(4) = 6 in both cases. Position (coordinates) changes but shape and size are invariant.

Answer

Side lengths, angles, and area are invariant; position changes.\text{Side lengths, angles, and area are invariant; position changes.}
Translation is a rigid motion (isometry) that preserves distances, angles, and area. The only thing that changes is the position of the figure. These preserved properties are called invariants of the transformation.

About Invariants Under Transformation

A property of a function is invariant under a transformation if it remains unchanged after the transformation is applied to the function.

Learn more about Invariants Under Transformation →

More Invariants Under Transformation Examples

Example 2 medium

A rectangle with vertices [formula], [formula], [formula], [formula] is dilated by a scale factor of

Example 3 medium

A figure is reflected over the [formula]-axis. Determine whether each property is invariant: (a) sid

Example 4 hard

Under a shear transformation defined by [formula], determine whether the area of a unit square with

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