Invariant

Invariant

A word describing of a property which can not be changed by a given transformation.

Two pentagons on either side of a vertical reflection line, labeled "pre-image" and "image," with equal areas showing area is...

See also

Reflection, pre-image, image

Key Formula

P(F') = P(F)

Where:

$P$ = A property (such as distance, angle measure, or area)
$F$ = The original figure (pre-image)
$F'$ = The transformed figure (image)

Worked Example

Problem: Triangle ABC has vertices A(1, 2), B(5, 2), and B(3, 6). The triangle is translated 4 units to the right. Show that the side lengths are invariant under this translation.

Step 1: Find the original side length AB using the distance formula.

AB = \sqrt{(5-1)^2 + (2-2)^2} = \sqrt{16} = 4

Step 2: Apply the translation (x, y) → (x + 4, y) to get the new vertices: A'(5, 2), B'(9, 2), C'(7, 6).

A'(1+4,\, 2) = A'(5,\, 2), \quad B'(5+4,\, 2) = B'(9,\, 2)

Step 3: Find the new side length A'B' using the distance formula.

A'B' = \sqrt{(9-5)^2 + (2-2)^2} = \sqrt{16} = 4

Step 4: Compare the two lengths. Since AB = A'B' = 4, the distance is invariant under this translation. You can verify the same holds for sides AC, BC, and all angle measures.

AB = A'B' = 4 \quad \checkmark

Answer: The side length AB = 4 is unchanged after translation, confirming that distance is an invariant of translations.

Another Example

This example differs because it involves a dilation (a non-rigid transformation) rather than a translation, showing that some properties are invariant while others are not — the key idea that invariance depends on which transformation is applied.

Problem: A rectangle with vertices P(0, 0), Q(6, 0), R(6, 3), S(0, 3) is dilated by a scale factor of 2 centered at the origin. Determine which properties are invariant and which are not.

Step 1: Apply the dilation (x, y) → (2x, 2y) to each vertex.

P'(0,0),\; Q'(12,0),\; R'(12,6),\; S'(0,6)

Step 2: Check angle measures. The original rectangle has four 90° angles. The image is also a rectangle with four 90° angles.

\angle P = \angle P' = 90°

Step 3: Check side lengths. The original PQ = 6; the new P'Q' = 12. Side lengths have changed, so distance is NOT invariant under dilation.

PQ = 6, \quad P'Q' = 12 \quad \Rightarrow \quad PQ \neq P'Q'

Step 4: Check the ratio of sides. Original ratio PQ : PS = 6 : 3 = 2 : 1. New ratio P'Q' : P'S' = 12 : 6 = 2 : 1. The ratio of corresponding sides is invariant.

\frac{PQ}{PS} = \frac{P'Q'}{P'S'} = 2

Answer: Under dilation, angle measures and the ratio of side lengths are invariant, but actual distances and area are not invariant.

Frequently Asked Questions

What is an invariant point in a transformation?

An invariant point is a specific point that maps to itself under a transformation — it does not move. For example, in a reflection over the x-axis, every point on the x-axis is an invariant point because reflecting (a, 0) gives (a, 0). Points not on the line of reflection will move and are therefore not invariant.

What properties are invariant under rigid transformations?

Rigid transformations (also called isometries) include translations, reflections, and rotations. Under any rigid transformation, distance, angle measure, parallelism, collinearity, and area are all invariant. The shape and size of the figure are completely preserved; only its position or orientation changes.

Is area invariant under dilation?

No. When you dilate a figure by scale factor k, the area is multiplied by k². For instance, dilating a triangle with area 10 by a factor of 3 produces a triangle with area 90. However, angle measures and the ratios of corresponding sides are invariant under dilation.

Rigid Transformation (Isometry) vs. Non-Rigid Transformation (e.g., Dilation)

	Rigid Transformation (Isometry)	Non-Rigid Transformation (e.g., Dilation)
Distance invariant?	Yes — all distances are preserved	No — distances are scaled by factor k
Angle measure invariant?	Yes	Yes
Area invariant?	Yes	No — area is multiplied by k²
Shape preserved?	Yes — congruent image	Yes — similar image (same shape, different size)
Examples	Translation, reflection, rotation	Dilation, stretches, compressions

Why It Matters

Understanding invariants is central to proving that two figures are congruent (all properties preserved by rigid transformations) or similar (angle measures and ratios preserved by dilations). In standardized geometry courses and exams, you are regularly asked to identify which properties a transformation preserves or changes. The concept also extends beyond geometry into algebra and higher mathematics, where invariants help classify structures and simplify problems.

Common Mistakes

Mistake: Assuming that if a property is invariant under one transformation, it must be invariant under all transformations.

Correction: Invariance is always stated with respect to a specific transformation. Distance is invariant under reflections but not under dilations. Always specify which transformation you are considering.

Mistake: Confusing an invariant point with an invariant property.

Correction: An invariant point is a single point that does not move (maps to itself). An invariant property is a measurement or characteristic (like angle size or area) that remains the same for the entire figure. These are two distinct concepts.

Related Terms

Transformations — The operations under which invariants are studied
Reflection — A rigid transformation preserving distance and angles
Pre-Image of a Transformation — The original figure before transformation is applied
Image of a Transformation — The resulting figure after transformation is applied
Isometry — A transformation where distance is invariant
Dilation — A non-rigid transformation where angles are invariant but distances are not
Congruent — Figures related by transformations preserving all metric invariants
Similar — Figures related by transformations preserving angle invariants