The Central Idea

A transformation changes the position, orientation, or size of a figure according to a precise rule.

A transformation-based proof asks:

\[ \boxed{\text{Can one figure be mapped onto another while preserving the required properties?}} \]

If the answer is yes, the transformation itself becomes the proof mechanism.

Transformation as a Function

A geometric transformation can be viewed as a function that maps each point \(P\) to an image point \(P'\).

\[ T:P\mapsto P'. \]

The entire figure is transformed by applying the same rule to every point.

Preimage and Image

The original figure is called the preimage.

The transformed figure is called the image.

If point \(A\) maps to \(A'\), we may write

\[ T(A)=A'. \]

Rigid Motions

A rigid motion, or isometry, preserves all distances.

The basic rigid motions are:

Compositions of rigid motions are also rigid motions.

What Rigid Motions Preserve

Rigid motions preserve:

Congruence through Rigid Motion

Two figures are congruent if a sequence of rigid motions maps one exactly onto the other.

Symbolically:

\[ \boxed{ F_1\cong F_2 \iff \text{a rigid motion maps }F_1\text{ onto }F_2. } \]

Why This Proves Congruence

Since rigid motions preserve every corresponding distance and angle measure, the image has the same shape and size as the preimage.

Therefore rigid-motion equivalence is a geometric definition of congruence.

Translation

A translation moves every point the same distance in the same direction.

In vector form,

\[ P'\;=\;P+\mathbf{v}. \]

Every point has the same displacement vector.

Coordinate Rule for Translation

Translation by vector

\[ \langle a,b\rangle \]

maps

\[ (x,y) \]

to

\[ \boxed{(x+a,y+b)}. \]

Translation Preserves Direction

If a line is translated to a distinct image line, the two lines are parallel.

Thus translation provides a transformation-based explanation for parallelism.

Translation and Congruent Segments

If translation maps segment \(\overline{AB}\) to \(\overline{A'B'}\), then

\[ \boxed{AB=A'B'}. \]

Distance is invariant under translation.

Rotation

A rotation turns a figure through a fixed angle around a fixed center.

If \(O\) is the center and \(P\) maps to \(P'\), then

\[ OP=OP' \]

and

\[ m\angle POP'=\theta, \]

where \(\theta\) is the rotation angle.

Rotation Preserves Distance

Every point remains the same distance from the center of rotation.

More generally, all pairwise distances in the figure are preserved.

Therefore rotations are rigid motions.

Coordinate Rotation by \(90^\circ\)

A counterclockwise \(90^\circ\) rotation about the origin maps

\[ (x,y) \]

to

\[ \boxed{(-y,x)}. \]

Coordinate Rotation by \(180^\circ\)

\[ \boxed{(x,y)\mapsto(-x,-y)}. \]

Every point is sent through the origin to the point directly opposite.

Coordinate Rotation by \(270^\circ\)

A counterclockwise \(270^\circ\) rotation about the origin maps

\[ (x,y) \]

to

\[ \boxed{(y,-x)}. \]

General Rotation

A rotation by angle \(\theta\) about the origin can be written

\[ \boxed{ \begin{aligned} x'&=x\cos\theta-y\sin\theta,\\ y'&=x\sin\theta+y\cos\theta. \end{aligned} } \]

Reflection

A reflection maps a figure across a line called the line of reflection.

The line of reflection is the perpendicular bisector of the segment joining each point to its image.

Reflection across the \(x\)-axis

\[ \boxed{(x,y)\mapsto(x,-y)}. \]

Reflection across the \(y\)-axis

\[ \boxed{(x,y)\mapsto(-x,y)}. \]

Reflection across \(y=x\)

\[ \boxed{(x,y)\mapsto(y,x)}. \]

Reflection across \(y=-x\)

\[ \boxed{(x,y)\mapsto(-y,-x)}. \]

Reflection Preserves Distance and Angle

A reflected figure has exactly the same side lengths and angle measures as the original.

Therefore reflection proves congruence even though orientation reverses.

Orientation

Translation and rotation preserve orientation.

Reflection reverses orientation.

Orientation is therefore not required for congruence.

Dilation

A dilation changes size while preserving shape.

Given center \(O\) and scale factor \(k>0\), point \(P\) maps to \(P'\) on ray \(\overrightarrow{OP}\) so that

\[ OP'=k\,OP. \]

Coordinate Dilation from the Origin

A dilation with center at the origin and scale factor \(k\) maps

\[ (x,y) \]

to

\[ \boxed{(kx,ky)}. \]

What Dilations Preserve

Dilations preserve:

They do not generally preserve absolute length.

How Length Changes under Dilation

If the scale factor is \(k\), then

\[ \boxed{A'B'=k\,AB}. \]

How Area Changes under Dilation

Areas scale by

\[ \boxed{k^2}. \]

How Volume Changes under Dilation

In three dimensions, volumes scale by

\[ \boxed{k^3}. \]

Similarity through Transformations

Two figures are similar if one can be mapped onto the other by a dilation followed or preceded by rigid motions.

\[ \boxed{ F_1\sim F_2 \iff \text{a similarity transformation maps }F_1\text{ onto }F_2. } \]

Congruence as a Special Similarity

A rigid motion is a similarity transformation with scale factor

\[ 1. \]

Thus congruence is the special case of similarity in which size does not change.

Composition of Transformations

Transformations can be applied in sequence.

If \(T_1\) acts first and \(T_2\) acts second, the composition is

\[ T_2\circ T_1. \]

Order Can Matter

In general,

\[ T_2\circ T_1 \ne T_1\circ T_2. \]

Rotating and then translating may produce a different final image from translating and then rotating.

Composition of Rigid Motions

A composition of rigid motions is still a rigid motion.

Therefore distance and angle measure remain preserved throughout the entire sequence.

Composition and Congruence Proof

If a sequence of translation, rotation, and reflection maps one triangle exactly onto another, the triangles are congruent.

There is no need to verify every side and angle separately.

Transformation Invariants

An invariant is a property that remains unchanged under a transformation.

Proofs often depend on identifying the right invariant.

Rigid-Motion Invariants

Under rigid motions, important invariants include:

Dilation Invariants

Under dilation, useful invariants include:

Not Everything Is Invariant

Under dilation with \(k\ne1\), absolute lengths and areas change.

Under reflection, orientation changes.

A proof must use properties actually preserved by the transformation in question.

Transformation Proof of Segment Congruence

Suppose a rotation maps

\[ A\mapsto C \]

and

\[ B\mapsto D. \]

Since rotation preserves distance,

\[ \boxed{AB=CD}. \]

Transformation Proof of Angle Congruence

If a rigid motion maps

\[ \angle ABC \]

onto

\[ \angle DEF, \]

then

\[ \boxed{\angle ABC\cong\angle DEF}. \]

Transformation Proof of Triangle Congruence

If a rigid motion maps vertices

\[ A\mapsto D,\qquad B\mapsto E,\qquad C\mapsto F, \]

then

\[ \boxed{\triangle ABC\cong\triangle DEF}. \]

Transformation Proof of Triangle Similarity

If a dilation of scale factor \(k\), possibly combined with rigid motions, maps

\[ \triangle ABC \]

onto

\[ \triangle DEF, \]

then

\[ \boxed{\triangle ABC\sim\triangle DEF}. \]

Transformations and Correspondence

A mapping makes correspondence explicit.

If

\[ A\mapsto D,\quad B\mapsto E,\quad C\mapsto F, \]

then the correct congruence or similarity statement follows in that order.

Reflection and Perpendicular Bisectors

If reflection across line \(\ell\) maps \(A\) to \(B\), then \(\ell\) is the perpendicular bisector of \(\overline{AB}\).

Every point on \(\ell\) is fixed by the reflection and is therefore equidistant from \(A\) and \(B\).

Reflection and Angle Bisectors

If reflection across ray \(\overrightarrow{BD}\) maps one side of \(\angle ABC\) onto the other, then

\[ \angle ABD\cong\angle DBC. \]

Thus \(\overrightarrow{BD}\) is the angle bisector.

Reflection as a Proof of Isosceles Structure

In a symmetric isosceles triangle, reflection across the axis through the apex maps one base vertex to the other.

Corresponding sides and angles are therefore congruent.

Rotation and Regular Polygons

A regular \(n\)-gon is mapped onto itself by rotation through

\[ \frac{360^\circ}{n} \]

about its center.

This rotational symmetry proves equality of repeated sides and angles.

Rotation and Congruent Sectors

Connecting the center of a regular polygon to every vertex creates congruent central triangles.

Rotation maps each one onto the next.

Translation and Parallelograms

A translation mapping

\[ A\mapsto D \]

and

\[ B\mapsto C \]

sends segment \(\overline{AB}\) to \(\overline{DC}\).

Therefore the two segments are congruent and parallel.

Transformation Characterization of a Parallelogram

Opposite sides of a parallelogram can be related by a translation.

The figure also has \(180^\circ\) rotational symmetry about the intersection of its diagonals.

Half-Turn Symmetry of a Parallelogram

A \(180^\circ\) rotation about the diagonal intersection maps:

\[ A\leftrightarrow C \]

and

\[ B\leftrightarrow D. \]

This transformation explains several parallelogram properties at once.

Opposite Sides from Half-Turn Symmetry

Because the half-turn maps \(\overline{AB}\) to \(\overline{CD}\),

\[ AB=CD. \]

Likewise,

\[ BC=DA. \]

Opposite Angles from Half-Turn Symmetry

The same rotation maps opposite angles to one another.

Therefore,

\[ \angle A\cong\angle C \]

and

\[ \angle B\cong\angle D. \]

Diagonals from Half-Turn Symmetry

The center of the \(180^\circ\) rotation is the midpoint of each diagonal.

Therefore the diagonals bisect each other.

Reflection Symmetry of a Rectangle

A rectangle has reflection symmetries through lines connecting midpoints of opposite sides.

It also has \(180^\circ\) rotational symmetry.

Reflection Symmetry of a Rhombus

The diagonals of a rhombus serve as reflection axes.

This transformation viewpoint explains why the diagonals bisect the vertex angles.

Square Symmetry

A square has both the reflection symmetries of a rectangle and those of a rhombus.

It also has rotations through

\[ 90^\circ,\quad180^\circ,\quad270^\circ \]

that map the square onto itself.

Transformation Groups

The set of all rigid motions that map a figure onto itself forms its symmetry group.

This idea connects elementary geometry to abstract algebra.

Symmetry as an Invariant Structure

A symmetric figure remains unchanged as a whole under certain nontrivial transformations.

The transformation moves individual points while preserving the figure as a set.

Transformation Proof of Parallelism

If translation maps line \(\ell\) to a distinct line \(m\), then

\[ \boxed{\ell\parallel m}. \]

Translation preserves direction.

Transformation Proof of Perpendicularity

A \(90^\circ\) rotation maps any line through the center to a perpendicular line.

Thus if a transformation is known to rotate one line by \(90^\circ\), the image line is perpendicular to the original.

Transformation Proof of Midpoint

If a \(180^\circ\) rotation about point \(M\) maps \(A\) to \(B\), then

\[ MA=MB \]

and \(A\), \(M\), \(B\) are collinear.

Therefore \(M\) is the midpoint of \(\overline{AB}\).

Transformation Proof of a Perpendicular Bisector

If reflection across line \(\ell\) swaps \(A\) and \(B\), then \(\ell\) is the perpendicular bisector of \(\overline{AB}\).

Reflection structure alone establishes both bisection and perpendicularity.

Transformation Proof of an Angle Bisector

If reflection across ray \(\overrightarrow{BD}\) swaps the rays \(\overrightarrow{BA}\) and \(\overrightarrow{BC}\), then

\[ \angle ABD\cong\angle DBC. \]

Therefore \(\overrightarrow{BD}\) bisects the angle.

Dilation and Parallel Lines

A dilation maps a line not passing through the center to a parallel line.

This provides another transformation-based route to parallelism.

Dilation of a Line through the Center

A line passing through the center of dilation maps onto itself.

Points move along that same line, but the line as a set remains unchanged.

Dilation and Similar Triangles

If a triangle is dilated by factor \(k\), all corresponding side lengths are multiplied by \(k\), while all corresponding angles remain equal.

Therefore the original and image triangles are similar.

Transformation Proof of AA Similarity

If two triangles have two corresponding equal angles, a suitable dilation and rigid motion can align one side and orientation.

The third angle is then determined, and the triangles match in shape.

This gives a transformation interpretation of AA similarity.

Transformation Proof of SSS Similarity

If corresponding sides are proportional with scale factor \(k\), dilate one triangle by \(k\).

The resulting side lengths match the other triangle.

SSS congruence then completes the mapping.

Transformation Proof of SAS Similarity

Dilate one triangle so the two proportional sides match the corresponding sides of the other.

The included angle remains unchanged under dilation.

SAS congruence then establishes the rigid-motion match.

Coordinate Transformations as Proof Tools

Coordinate rules can verify a transformation point by point.

If every vertex of one polygon maps to the corresponding vertex of another under one transformation, the transformation-based conclusion follows.

Example: Translation Proof in Coordinates

Suppose

\[ A(1,2),\quad B(4,5) \]

and

\[ A'(6,0),\quad B'(9,3). \]

Both points move by

\[ \langle5,-2\rangle. \]

Therefore one segment is the translation image of the other, so the segments are congruent and parallel.

Example: Rotation Proof in Coordinates

Let

\[ A(2,1) \]

map to

\[ A'(-1,2). \]

This matches the \(90^\circ\) counterclockwise rule

\[ (x,y)\mapsto(-y,x). \]

If all vertices follow the same rule, the figures are congruent.

Example: Reflection Proof in Coordinates

Suppose

\[ A(3,4)\mapsto A'(-3,4). \]

This is reflection across the \(y\)-axis.

If every vertex follows

\[ (x,y)\mapsto(-x,y), \]

the image is congruent to the preimage.

Example: Dilation Proof in Coordinates

Suppose

\[ A(2,3)\mapsto A'(6,9) \]

and every vertex is multiplied by \(3\).

This is a dilation centered at the origin with scale factor

\[ 3. \]

The figures are similar.

Transformation-Based Proof and Auxiliary Structure

Sometimes a useful transformation is not obvious until a center, axis, or correspondence is identified.

Constructing a midpoint, perpendicular bisector, or symmetry line can reveal the appropriate transformation.

Finding a Rotation Center

If \(A\mapsto A'\) and \(B\mapsto B'\) under a rotation, the center lies on the perpendicular bisectors of

\[ \overline{AA'} \]

and

\[ \overline{BB'}. \]

Their intersection identifies the rotation center when the configuration is nondegenerate.

Finding a Reflection Axis

If reflection maps \(A\) to \(A'\), the reflection axis is the perpendicular bisector of

\[ \overline{AA'}. \]

With several corresponding point pairs, the same axis must serve all pairs.

Finding a Translation Vector

If \(A\mapsto A'\), the translation vector is

\[ \overrightarrow{AA'}. \]

Every other corresponding point pair must have the same displacement vector.

Finding a Dilation Center

For a dilation, the center lies on every line joining a point to its image.

Thus lines

\[ \overleftrightarrow{AA'} \]

and

\[ \overleftrightarrow{BB'} \]

meet at the dilation center when the geometry is nondegenerate.

Fixed Points

A fixed point satisfies

\[ T(P)=P. \]

Different transformations have different fixed-point structures.

Fixed Points of a Rotation

A nontrivial rotation in the plane has exactly one fixed point:

\[ \boxed{\text{the center of rotation}}. \]

Fixed Points of a Reflection

Every point on the reflection axis is fixed.

Points off the axis are exchanged with mirror-image points.

Fixed Points of a Translation

A nonzero translation has no fixed points.

Every point moves by the same nonzero vector.

Fixed Points of a Dilation

For a dilation with scale factor \(k\ne1\), the center of dilation is the unique fixed point.

Transformation Proof and Loci

Transformations can explain locus theorems.

A perpendicular bisector is the set of fixed points of the reflection that exchanges two endpoints.

An angle bisector is the set of fixed points of the reflection exchanging the two sides of the angle.

Transformation Proof and Constructions

Classical constructions often create the centers or axes of useful transformations.

Perpendicular bisectors identify reflection axes and circumcenters.

Angle bisectors identify symmetry directions.

Transformation Proof and Regular Figures

Regular polygons are especially natural in transformation language because their structure is defined by repeated rotational and reflection symmetries.

Transformation Proof Is Often More Global

A side-angle proof may compare several individual parts.

A transformation proof can establish many relationships at once by showing that the entire figure maps onto another.

Transformation Proof Is Not Always Shorter

If the required transformation is difficult to identify, a synthetic or coordinate proof may be more efficient.

The best method depends on the structure of the problem.

Common Error: Calling Every Movement a Translation

In a translation, every point moves by exactly the same vector.

If directions differ among points, the transformation is not a translation.

Common Error: Ignoring the Center of Rotation

Rotation requires a fixed center.

Equal angle turning without a common center does not define one rotation of the entire figure.

Common Error: Ignoring the Reflection Axis

A reflection axis must be the perpendicular bisector of every segment joining a point to its image.

Visual left-right resemblance alone is not sufficient.

Common Error: Treating Dilation as a Rigid Motion

A dilation with

\[ k\ne1 \]

changes lengths.

It proves similarity, not congruence.

Common Error: Scaling Angle Measures

Dilation changes lengths but preserves angle measures.

An angle measuring

\[ 40^\circ \]

remains

\[ 40^\circ \]

after dilation.

Common Error: Forgetting Orientation Reversal

Reflection preserves congruence but reverses orientation.

Clockwise vertex order may become counterclockwise.

Common Error: Applying Different Rules to Different Vertices

One transformation-based proof requires one consistent transformation or one clearly stated composition applied to the entire figure.

Arbitrarily choosing separate rules for separate points does not establish a single mapping.

Common Error: Ignoring Correspondence

A transformation tells exactly which vertex maps to which.

The congruence or similarity statement must respect that mapping order.

Common Error: Assuming a Transformation from Appearance

Two figures may look like rotated or reflected copies without satisfying one exact transformation rule.

Verify the mapping mathematically.

Common Error: Misordering a Composition

If a rotation occurs first and a translation second, the composition must be applied in that order.

Reversing the order may produce a different image.

Common Error: Using a Noninvariant Property

A proof should rely only on properties preserved by the transformation.

For example, absolute length is not invariant under a nontrivial dilation.

Common Error: Proving Only Some Vertices Map

To prove one polygon maps onto another, the transformation must send all relevant vertices and edges to the corresponding image structure.

Checking only one point pair is usually insufficient.

A Reliable Rigid-Motion Proof Workflow

  1. Identify the two figures and their intended correspondence.
  2. Look for translation, rotation, reflection, or a composition.
  3. Identify any required vector, center, angle, or axis.
  4. Verify that the same transformation maps every key point correctly.
  5. Invoke rigid-motion invariants.
  6. Conclude congruence or the required geometric property.

A Reliable Similarity-Transformation Workflow

  1. Identify corresponding vertices.
  2. Determine the scale factor.
  3. Identify the dilation center when needed.
  4. Apply any necessary rigid motions to align orientation and position.
  5. Verify the complete mapping.
  6. Conclude similarity and use the scale factor for corresponding lengths.

A Reliable Symmetry-Proof Workflow

  1. Identify the proposed symmetry transformation.
  2. Find its center, axis, or angle.
  3. Track representative points or vertices.
  4. Verify the figure maps onto itself.
  5. Use the symmetry to conclude equal sides, equal angles, midpoint properties, or other invariants.

Example: Translation Proves Opposite Sides Congruent

Suppose translation by vector \(\mathbf{v}\) maps

\[ A\mapsto D \]

and

\[ B\mapsto C. \]

Then segment \(\overline{AB}\) maps to \(\overline{DC}\).

Since translation preserves length,

\[ \boxed{AB=DC}. \]

Example: Reflection Proves Base Angles Congruent

Suppose reflection across line \(\ell\) fixes vertex \(A\) and maps

\[ B\leftrightarrow C. \]

Then

\[ AB=AC \]

and the base angles map to one another.

Therefore,

\[ \boxed{\angle B\cong\angle C}. \]

Example: Rotation Proves Repeated Congruence

In a regular hexagon, a \(60^\circ\) rotation about the center maps each side to the next.

Therefore all six sides are congruent.

Example: Dilation Proves Similarity

Suppose a dilation of scale factor

\[ 2 \]

maps \(\triangle ABC\) onto \(\triangle A'B'C'\).

Then

\[ A'B'=2AB, \] \[ B'C'=2BC, \] \[ A'C'=2AC, \]

while all corresponding angles remain congruent.

Therefore the triangles are similar.

Example: Half-Turn Proves Diagonal Bisection

Suppose a \(180^\circ\) rotation about point \(M\) maps

\[ A\leftrightarrow C \]

and

\[ B\leftrightarrow D. \]

Then

\[ MA=MC \]

and

\[ MB=MD. \]

Therefore \(M\) is the midpoint of both diagonals.

Example: Reflection Proves a Perpendicular Bisector

Suppose reflection across line \(\ell\) maps

\[ A\mapsto B. \]

Then \(\ell\) is the perpendicular bisector of \(\overline{AB}\).

Therefore every point \(P\) on \(\ell\) satisfies

\[ \boxed{PA=PB}. \]

Practice 1: Rigid Motion

Which basic transformations are rigid motions?

Practice 2: Translation Rule

Translate

\[ (3,-2) \]

by vector

\[ \langle4,5\rangle. \]

Find the image point.

Practice 3: \(90^\circ\) Rotation

Rotate

\[ (2,5) \]

\(90^\circ\) counterclockwise about the origin.

Practice 4: Reflection

Reflect

\[ (-3,7) \]

across the \(y\)-axis.

Practice 5: Dilation

Dilate

\[ (4,-2) \]

from the origin by scale factor

\[ \frac12. \]

Practice 6: Congruence

A sequence of rigid motions maps one triangle exactly onto another.

What relationship holds between the triangles?

Practice 7: Similarity

A dilation followed by a rotation maps one triangle exactly onto another.

What relationship holds between the triangles?

Practice 8: Reflection Axis

Reflection maps point \(A\) to point \(B\).

What geometric relationship does the reflection axis have to \(\overline{AB}\)?

Practice 9: Half-Turn Symmetry

A \(180^\circ\) rotation about \(M\) maps \(A\) to \(C\).

What length relationship follows?

Practice 10: Identify the Error

A student says, “The figure was dilated by scale factor \(3\), so all its angles became three times as large.”

Explain the error.

Practice Answers

Practice 1

The basic rigid motions are

\[ \boxed{\text{translations, rotations, and reflections}}. \]

Practice 2

Add the translation vector:

\[ (3,-2)+(4,5)=(7,3). \]

Therefore,

\[ \boxed{(7,3)}. \]

Practice 3

Use

\[ (x,y)\mapsto(-y,x). \]

Thus

\[ (2,5)\mapsto\boxed{(-5,2)}. \]

Practice 4

Reflection across the \(y\)-axis uses

\[ (x,y)\mapsto(-x,y). \]

Therefore,

\[ (-3,7)\mapsto\boxed{(3,7)}. \]

Practice 5

Multiply both coordinates by

\[ \frac12: \] \[ (4,-2)\mapsto\boxed{(2,-1)}. \]

Practice 6

Rigid motions preserve shape and size.

Therefore,

\[ \boxed{\text{the triangles are congruent}}. \]

Practice 7

A dilation combined with a rigid motion is a similarity transformation.

Therefore,

\[ \boxed{\text{the triangles are similar}}. \]

Practice 8

The reflection axis is

\[ \boxed{\text{the perpendicular bisector of }\overline{AB}}. \]

Practice 9

Rotation preserves distance from its center, so

\[ \boxed{MA=MC}. \]

Practice 10

Dilations preserve angle measure.

Scale factor \(3\) triples corresponding lengths, not angle measures.

Each angle remains exactly the same measure.

What to Remember

Transformation-based proof establishes geometric relationships by identifying exact mappings and the properties those mappings preserve.

The essential habit is: identify the transformation, verify the complete mapping, then use only properties that are invariant under that transformation.

Where We Go Next

Transformation-based proof shows how geometric structure persists under carefully defined mappings. The final lesson in this sequence turns attention to what happens when a proposed theorem fails or when a familiar configuration collapses into a boundary case.

The next lesson, Counterexamples and Degenerate Cases, develops methods for testing universal claims, exposing hidden assumptions, analyzing limiting configurations, and deciding when special cases must be excluded or treated separately.