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: An invariant is any property of a figure that a given transformation leaves unchanged.

Common stuck point: The procedure for geometric invariance is the easy part; the trap is assuming all transformations preserve length. Asking "Am I asking which property a transformation leaves unchanged, not where the figure moves?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I asking which property a transformation leaves unchanged, not where the figure moves?

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?

Answer

Invariant: (a) side lengths and (b) angle measures. Not invariant: (c) orientation, (d)

x

-coordinates.

First step

Step 1: Both reflection and rotation are isometries (distance-preserving), so side lengths are invariant. ✓

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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.

Example 3

medium

A triangle with area

12

is dilated by scale factor

k = 4

about a point. Find the area of the image.

Example 4

medium

A rectangle has dimensions

8 \times 5

. After a dilation by scale factor

k = \tfrac{1}{2}

, find the ratio (new perimeter)/(original perimeter) and the ratio (new area)/(original area).

Example 5

medium

A triangle has area

20

and is reflected across a line, then dilated by scale factor

3

. Compute the area of the final image.

Example 6

hard

A rectangle's perimeter is

40

. After a dilation, the image rectangle has perimeter

100

. Find the ratio of areas (image:original).

Example 7

hard

A figure has 4-fold rotational symmetry about a point

P

. Explain why any quantity preserved by rotation about

P

is automatically the same for all four rotated copies.

Example 8

challenge

Klein's Erlangen program defines a geometry by a group of transformations and asks which properties are invariant. Order these geometries from most invariants to fewest: Euclidean, Projective, Similarity, Affine.

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.

Example 3

easy

Is a figure's area invariant (unchanged) under a translation?

Example 4

easy

Is length invariant under a dilation (factor not 1)?

Example 5

easy

Are angle measures invariant under rotation?

Example 6

easy

Which transformation does NOT leave area invariant: rotation, reflection, or dilation?

Example 7

easy

What is an 'invariant' in geometry?

Example 8

easy

Is the number of sides of a polygon invariant under any rigid motion?

Example 9

easy

Is orientation invariant under a reflection?

Example 10

easy

Are angles invariant under dilation?

Example 11

medium

Which property is invariant under ALL four basic transformations (translation, rotation, reflection, dilation)?

Example 12

medium

A property is invariant under translation and rotation but NOT reflection. Name such a property.

Example 13

medium

Is the ratio of two side lengths of a figure invariant under dilation?

Example 14

medium

Is collinearity (points lying on a line) invariant under all four basic transformations?

Example 15

medium

Why is area NOT invariant under dilation, but IS invariant under reflection?

Example 16

medium

Is betweenness (one point lying between two others on a line) invariant under rigid motions?

Example 17

medium

Under rotation, a point is 5 units from the center. Name an invariant this illustrates.

Example 18

medium

Two figures are congruent. Which is true: they share all rigid-motion invariants, or all dilation invariants?

Example 19

challenge

Explain why 'similar figures' are exactly those sharing all the invariants of the group generated by rigid motions plus dilations.

Example 20

challenge

A figure is translated, rotated, and reflected in sequence. Which of {length, angle, area, orientation} are invariant overall?

Example 21

challenge

Why is the sum of a triangle's interior angles (

180^\circ

) an invariant under every transformation we've studied?

Example 22

challenge

How does the idea of invariance let you prove two figures are NOT congruent without measuring everything?

Example 23

easy

Under a translation by

\langle 4, -7\rangle

, is the perimeter of a triangle invariant?

Example 24

easy

A square has side length

6

. After a dilation centered at the origin with scale factor

k = 3

, what is the new side length, and is side length invariant?

Example 25

easy

A triangle has angles

40^\circ, 60^\circ, 80^\circ

. After a dilation by scale factor

k = 0.5

, list the three angles of the image.

Example 26

easy

True or False: the slope of a line is invariant under translation.

Example 27

medium

A circle of radius

5

is rotated

90^\circ

counterclockwise about a point

20

units away. Which of {radius, center coordinates, area, circumference} are invariant?

Example 28

medium

A polygon has vertices

A,B,C,D

in clockwise order. After a single reflection, are the image vertices in clockwise or counterclockwise order?

Example 29

medium

A triangle has interior angles

30^\circ, 60^\circ, 90^\circ

. After three rigid motions in succession, list its interior angles.

Example 30

medium

Under a shear

(x,y) \mapsto (x+2y, y)

, is the area of the unit square invariant?

Example 31

medium

A right triangle with legs

3

and

4

is reflected, then translated. Find the length of its hypotenuse after the transformations.

Example 32

medium

Two figures are similar with ratio

2:3

. If a length in the first measures

10

, find the corresponding length in the second.

Example 33

hard

A triangle has vertices

(0,0),(6,0),(0,8)

. After a dilation centered at

(0,0)

with factor

k

, the image has area

54

. Find

|k|

Example 34

hard

A figure is reflected, rotated, then dilated by factor

2

. If the original area is

A

, find the final area in terms of

A

Example 35

hard

In the plane, three points

A,B,C

are collinear. Are they still collinear after an affine transformation? Justify in one sentence.

Example 36

hard

A unit square is mapped by the linear transformation with matrix

\begin{pmatrix}3 & 1\\ 0 & 2\end{pmatrix}

. Find the area of the image.

Example 37

hard

Triangle

T_1

has sides

5,12,13

; triangle

T_2

has sides

10,24,26

. List two invariants they share under similarity (not congruence).

Example 38

challenge

Four collinear points lie at

0,2,5,10

on a number line. Compute the cross-ratio

(AC\cdot BD)/(BC\cdot AD)

where

A=0,B=2,C=5,D=10

. (This value is invariant under projective transformations.)

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Related Concepts

Geometric Transformation Congruence Similarity

Background Knowledge

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

transformation geo