Transformations and Coordinate Geometry

A transformation changes the position, orientation, or size of a geometric figure. On a coordinate plane, transformations can be described precisely by rules that change the coordinates of each point.

The four major transformations are:

• translation — slides a figure
• reflection — flips a figure
• rotation — turns a figure
• dilation — changes the size of a figure

Transformations are important in geometry because they allow us to investigate congruence, similarity, symmetry, orientation, and geometric relationships.

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Preimage and Image

The original figure before a transformation is called the preimage.

The transformed figure is called the image.

If point A is transformed, its image is usually written:

A′

read as:

A prime

For example:

A(2, 3) → A′(7, 5)

If an entire triangle ABC is transformed, its image may be named:

A′B′C′

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Coordinate Rules

A transformation can be represented using a coordinate rule.

For example:

(x, y) → (x + 4, y − 2)

means:

• add 4 to every x-coordinate
• subtract 2 from every y-coordinate

This transformation moves every point:

4 units right and 2 units down

Coordinate rules allow transformations to be performed without relying only on a diagram.

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Translations

A translation slides every point of a figure the same distance in the same direction.

A translation does not:

• turn the figure
• flip the figure
• change its size
• change its shape

For a translation:

(x, y) → (x + a, y + b)

where:

• a controls horizontal movement
• b controls vertical movement

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Horizontal Translations

If:

a > 0

the figure moves right.

If:

a < 0

the figure moves left.

Examples:

(x, y) → (x + 5, y)

means 5 units right.

(x, y) → (x − 3, y)

means 3 units left.

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Vertical Translations

If:

b > 0

the figure moves up.

If:

b < 0

the figure moves down.

Examples:

(x, y) → (x, y + 6)

means 6 units up.

(x, y) → (x, y − 4)

means 4 units down.

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Translation Example

Triangle ABC has:

A(1, 2)

B(5, 2)

C(3, 6)

Translate the triangle:

4 units right and 3 units down

The rule is:

(x, y) → (x + 4, y − 3)

Apply it to each point.

A(1, 2):

A′ = (1 + 4, 2 − 3)

A′(5, −1)

B(5, 2):

B′(9, −1)

C(3, 6):

C′(7, 3)

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Translation Vectors

A translation can also be represented by a vector.

For example:

(4, −3)

means:

• move 4 units in the positive x-direction
• move 3 units in the negative y-direction

Therefore:

(x, y) → (x + 4, y − 3)

Translations are sometimes described as adding the same vector to every point.

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Finding a Translation Rule

Suppose:

A(2, 5) → A′(8, 1)

Compare coordinates.

Horizontal change:

8 − 2 = 6

Vertical change:

1 − 5 = −4

Therefore, the translation is:

6 units right and 4 units down

Rule:

(x, y) → (x + 6, y − 4)

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Reflections

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

The original figure and its image appear on opposite sides of this line.

Each point and its image are:

• the same perpendicular distance from the line of reflection
• positioned on opposite sides of the line

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Reflection Across the x-Axis

For reflection across the x-axis:

(x, y) → (x, −y)

The x-coordinate stays the same.

The sign of the y-coordinate changes.

Example:

A(4, 3)

becomes:

A′(4, −3)

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Reflection Across the y-Axis

For reflection across the y-axis:

(x, y) → (−x, y)

The y-coordinate stays the same.

The sign of the x-coordinate changes.

Example:

B(5, −2)

becomes:

B′(−5, −2)

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Reflection Across y = x

For reflection across:

y = x

the coordinates switch places:

(x, y) → (y, x)

Example:

C(3, 7)

becomes:

C′(7, 3)

Another example:

D(−2, 5)

becomes:

D′(5, −2)

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Reflection Across y = −x

For reflection across:

y = −x

switch the coordinates and change both signs:

(x, y) → (−y, −x)

Example:

A(2, 5)

becomes:

A′(−5, −2)

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Reflection Across a Vertical Line

For reflection across:

x = a

the y-coordinate remains unchanged.

The point moves to the opposite side of x = a at the same perpendicular distance.

For example, reflect:

A(2, 4)

across:

x = 5

A is:

5 − 2 = 3 units

left of the reflection line.

Therefore, A′ must be 3 units to the right:

x = 5 + 3

x = 8

Thus:

A′(8, 4)

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Reflection Across a Horizontal Line

Suppose:

A(3, 7)

is reflected across:

y = 4

The point is:

7 − 4 = 3 units

above the line.

Its image must be 3 units below:

4 − 3 = 1

Therefore:

A′(3, 1)

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General Reflection Rules

For reflection across:

x = a

the rule is:

(x, y) → (2a − x, y)

For reflection across:

y = b

the rule is:

(x, y) → (x, 2b − y)

These rules work for any vertical or horizontal line of reflection.

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Rotations

A rotation turns a figure around a fixed point called the center of rotation.

Unless stated otherwise, coordinate rotation rules usually refer to rotations around:

the origin (0, 0)

Rotations can be:

• clockwise
• counterclockwise

and are measured in degrees.

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90° Counterclockwise Rotation

The rule is:

(x, y) → (−y, x)

Example:

A(4, 2)

becomes:

A′(−2, 4)

Notice that:

• coordinates switch places
• the original y-coordinate changes sign

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90° Clockwise Rotation

The rule is:

(x, y) → (y, −x)

Example:

A(4, 2)

becomes:

A′(2, −4)

A 90° clockwise rotation is equivalent to a:

270° counterclockwise rotation

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180° Rotation

The rule is:

(x, y) → (−x, −y)

This rule is the same whether the rotation is clockwise or counterclockwise.

Example:

A(4, 2)

becomes:

A′(−4, −2)

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270° Counterclockwise Rotation

The rule is:

(x, y) → (y, −x)

This is equivalent to a 90° clockwise rotation.

For example:

A(−3, 5)

becomes:

A′(5, 3)

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Rotation Rules Summary

These rules are worth becoming familiar with.

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Rotation Example

Triangle ABC has:

A(1, 2)

B(5, 2)

C(3, 6)

Rotate the triangle:

90° counterclockwise about the origin

Use:

(x, y) → (−y, x)

A(1, 2):

A′(−2, 1)

B(5, 2):

B′(−2, 5)

C(3, 6):

C′(−6, 3)

The triangle has changed orientation and position but not size or shape.

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Understanding Rotation Geometrically

During a rotation, every point remains the same distance from the center of rotation.

For example, if point A is:

5 units from the origin

before rotation, then A′ is:

5 units from the origin

after rotation.

This provides a useful way to check whether a rotation has been performed correctly.

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Dilations

A dilation changes the size of a figure while preserving its shape.

Dilations use a:

center of dilation

and a:

scale factor

When the center of dilation is the origin, the rule is:

(x, y) → (kx, ky)

where k is the scale factor.

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Enlargement

If:

k > 1

the image is larger than the preimage.

Example:

k = 2

A(3, 4)

becomes:

A′(6, 8)

because:

(x, y) → (2x, 2y)

Every distance from the center of dilation is doubled.

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Reduction

If:

0 < k < 1

the image becomes smaller.

Example:

k = 1/2

A(6, 8)

becomes:

A′(3, 4)

The new figure has the same shape but smaller dimensions.

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Dilation Example

Triangle ABC has:

A(2, 2)

B(6, 2)

C(4, 6)

Dilate it about the origin using:

k = 2

Then:

A′ = (4, 4)

B′ = (12, 4)

C′ = (8, 12)

Every side length becomes:

2 times the original length

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Effects of Dilation on Length

If the scale factor is k:

new length = k × original length

Suppose a triangle has a side length of:

8 cm

and is dilated by:

k = 3

The corresponding image side is:

3 × 8

24 cm

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Effects of Dilation on Perimeter

Because every side length is multiplied by k:

new perimeter = k × original perimeter

If:

P = 20 cm

and:

k = 1.5

then:

P′ = 1.5(20)

P′ = 30 cm

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Effects of Dilation on Area

Area changes differently from length.

If the scale factor is k:

new area = k² × original area

Suppose:

original area = 12 cm²

and:

k = 3

Then:

new area = 3²(12)

new area = 9(12)

new area = 108 cm²

This occurs because both dimensions are multiplied by the scale factor.

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Rigid Transformations

Translations, reflections, and rotations are called rigid transformations or isometries.

They preserve:

• side lengths
• angle measures
• area
• perimeter
• parallel relationships
• perpendicular relationships
• overall size and shape

Therefore, the image is:

congruent to the preimage

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What Reflections Change

Although a reflection preserves lengths and angles, it reverses the figure's orientation.

Suppose the vertices of a triangle appear in clockwise order:

A → B → C

After reflection, the corresponding vertices may appear counterclockwise:

A′ → B′ → C′

The figure remains congruent, but its orientation has reversed.

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What Rotations Change

A rotation changes:

• position
• direction of individual sides
• orientation relative to the coordinate axes

But it preserves:

• lengths
• angles
• area
• shape
• clockwise/counterclockwise vertex order

Therefore, rotation does not create a mirror image.

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What Translations Change

A translation changes the figure's:

• location

It does not change:

• size
• shape
• orientation
• side direction relative to parallel copies
• angle measures

Every point moves by the same translation vector.

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What Dilations Preserve

Dilations preserve:

• angle measures
• overall shape
• parallel relationships

But they do not generally preserve:

• lengths
• perimeter
• area

Therefore, a dilation creates a figure that is:

similar to the original

but not necessarily congruent.

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Congruence and Similarity

Two figures are congruent if they have the same size and shape.

Translations, reflections, and rotations produce congruent images.

Two figures are similar if they have the same shape and proportional dimensions.

Dilations produce similar figures.

If the dilation scale factor is 1, the image is also congruent to the original.

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Checking Length Preservation

Suppose:

A(1, 2)

B(5, 5)

Distance AB:

AB = √[(5 − 1)² + (5 − 2)²]

AB = √(16 + 9)

AB = 5

Translate both points 3 units right:

A′(4, 2)

B′(8, 5)

Now:

A′B′ = √[(8 − 4)² + (5 − 2)²]

A′B′ = √25

A′B′ = 5

The distance is unchanged.

This confirms that translation preserves length.

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Checking a Dilation

Suppose:

A(1, 1)

B(4, 5)

Then:

AB = √[(4 − 1)² + (5 − 1)²]

AB = √25

AB = 5

Dilate by:

k = 2

Then:

A′(2, 2)

B′(8, 10)

Distance:

A′B′ = √[(8 − 2)² + (10 − 2)²]

A′B′ = √(36 + 64)

A′B′ = 10

Therefore:

A′B′ = 2(AB)

which matches the scale factor.

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Symmetry

A figure has symmetry if a transformation maps the figure onto itself.

Two important types are:

• reflectional symmetry
• rotational symmetry

Coordinate geometry can help verify both types.

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Reflectional Symmetry

A figure has reflectional symmetry if it can be reflected across a line and remain unchanged.

The line is called a:

line of symmetry

Examples include:

• a square
• a rectangle
• an isosceles triangle
• some regular polygons

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Coordinate Test for Reflectional Symmetry

Suppose a figure contains:

A(3, 2)

and:

B(−3, 2)

These points are reflections across the y-axis because:

(x, y) → (−x, y)

If every point of the figure has a matching reflected point, the figure may have symmetry about:

the y-axis

Similar coordinate checks can be made for:

• x-axis
• y-axis
• y = x
• y = −x
• other vertical or horizontal lines

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Rotational Symmetry

A figure has rotational symmetry if it can be rotated through an angle less than 360° and map onto itself.

For example, a square maps onto itself after:

• 90°
• 180°
• 270°
• 360°

A non-square rectangle maps onto itself after:

• 180°
• 360°

The smallest positive rotation that maps a figure onto itself is called its:

angle of rotational symmetry

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Order of Rotational Symmetry

The order tells us how many times a figure maps onto itself during one full 360° turn.

For a square:

order = 4

because:

360°/4 = 90°

For an equilateral triangle:

order = 3

because:

360°/3 = 120°

For a regular hexagon:

order = 6

because:

360°/6 = 60°

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Multiple Transformations

Figures can undergo more than one transformation.

For example:

1. translate
2. reflect
3. rotate
4. dilate

This sequence is called a composition of transformations.

The transformations must be performed in the specified order.

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Why Order Matters

Suppose point:

A(2, 1)

is first translated:

3 units right

Then reflected across the y-axis.

First:

A(2, 1) → (5, 1)

Then reflect:

(5, 1) → (−5, 1)

Final position:

(−5, 1)

Now reverse the order.

First reflect:

A(2, 1) → (−2, 1)

Then translate right 3:

(−2, 1) → (1, 1)

The final positions are different.

Therefore:

the order of transformations can matter.

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Worked Multiple-Transformation Example

Triangle ABC has:

A(1, 1)

B(4, 1)

C(2, 3)

First translate:

2 units right and 1 unit up

Translation rule:

(x, y) → (x + 2, y + 1)

This gives:

A′(3, 2)

B′(6, 2)

C′(4, 4)

Now reflect across the x-axis:

(x, y) → (x, −y)

Final coordinates:

A″(3, −2)

B″(6, −2)

C″(4, −4)

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Prime Notation in Multiple Transformations

Prime notation can help track stages.

Original:

A

After first transformation:

A′

After second:

A″

After third:

A‴

This prevents confusion when several coordinate rules are applied.

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Combining Two Translations

Suppose a figure is translated:

3 units right and 2 units up

then:

5 units left and 4 units up.

Horizontal movement:

3 − 5 = −2

Vertical movement:

2 + 4 = 6

Therefore, the two translations are equivalent to one translation:

2 units left and 6 units up

Rule:

(x, y) → (x − 2, y + 6)

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Two Reflections Across the Same Line

If a figure is reflected across a line and then reflected across the same line again, it returns to its original position.

For example:

A(3, 4)

Reflect across y-axis:

A′(−3, 4)

Reflect across y-axis again:

A″(3, 4)

Therefore:

two reflections across the same line undo each other.

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Two 90° Rotations

Two successive 90° counterclockwise rotations are equivalent to:

one 180° rotation

For:

A(3, 2)

First rotation:

(3, 2) → (−2, 3)

Second rotation:

(−2, 3) → (−3, −2)

This matches the 180° rule:

(x, y) → (−x, −y)

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Inverse Transformations

An inverse transformation reverses another transformation.

Examples:

Translation right 5:

inverse = translation left 5

90° clockwise rotation:

inverse = 90° counterclockwise rotation

Reflection:

its own inverse

Dilation by k:

inverse dilation has scale factor:

1/k

provided k ≠ 0.

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Finding an Original Point

Suppose a point is translated:

4 units right and 3 units down

and its image is:

A′(9, 2)

To recover A, use the inverse:

4 units left and 3 units up.

A = (9 − 4, 2 + 3)

A(5, 5)

Inverse transformations allow us to work backward.

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Transformations of Lines

Transformations also affect lines.

Translations preserve slope.

A line with slope:

m = 3

will have a translated image with slope:

m = 3

The two distinct lines are therefore parallel.

Rotations and reflections may change the numerical slope, but they preserve geometric relationships such as angles.

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Transformations of Parallel Lines

Rigid transformations preserve parallelism.

If:

AB ∥ CD

before a translation, rotation, or reflection, then:

A′B′ ∥ C′D′

after the transformation.

Dilations also preserve parallel relationships.

Therefore, all four major transformations preserve parallelism.

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Transformations of Perpendicular Lines

Suppose:

AB ⟂ BC

A rigid transformation preserves the 90° angle.

Therefore:

A′B′ ⟂ B′C′

Dilations also preserve angle measures.

Thus perpendicularity is preserved by all four major transformations.

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Transformations and Area

Translations:

area unchanged

Reflections:

area unchanged

Rotations:

area unchanged

Dilations:

area multiplied by k²

For example, if a polygon has area:

20 cm²

and is dilated by:

k = 2.5

then:

new area = 2.5²(20)

new area = 6.25(20)

new area = 125 cm²

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Transformations and Perimeter

Translations, reflections, and rotations preserve perimeter.

Dilations change perimeter by the scale factor.

If:

P = 36 cm

and:

k = 1/2

then:

P′ = 1/2(36)

P′ = 18 cm

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Coordinate Transformations in Design

Transformations are widely used in:

• architecture
• engineering
• computer graphics
• animation
• video games
• manufacturing
• mapmaking
• pattern design
• robotics

For example, a computer can create repeated objects by translating an original design rather than redrawing it.

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Computer Graphics

In computer graphics, objects are represented using coordinates.

Transformations can:

• move objects
• rotate objects
• flip images
• resize objects

For example, a game character might be translated across the screen while a rotating wheel undergoes continuous rotations.

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Architecture and Pattern Design

Transformations are useful for creating repeating patterns.

A tile design might be:

• translated repeatedly
• reflected across boundaries
• rotated around points

These transformations can create complex designs from a relatively simple starting shape.

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Robotics

A robot's position may be represented by coordinates.

Moving a robot can involve:

• translation to a new location
• rotation to face a new direction

Robotics therefore combines geometric transformations with measurements of position and orientation.

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A Complete Translation and Reflection Problem

Rectangle ABCD has:

A(1, 1)

B(5, 1)

C(5, 4)

D(1, 4)

First translate:

3 units right and 2 units up

Then reflect across the y-axis.

After translation:

A′(4, 3)

B′(8, 3)

C′(8, 6)

D′(4, 6)

After reflection:

A″(−4, 3)

B″(−8, 3)

C″(−8, 6)

D″(−4, 6)

The rectangle remains congruent to the original because both transformations are rigid.

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A Complete Rotation and Dilation Problem

Triangle ABC has:

A(1, 2)

B(4, 2)

C(2, 5)

First rotate 90° counterclockwise about the origin.

Rule:

(x, y) → (−y, x)

This gives:

A′(−2, 1)

B′(−2, 4)

C′(−5, 2)

Now dilate by:

k = 2

This gives:

A″(−4, 2)

B″(−4, 8)

C″(−10, 4)

The final figure has the same shape as the original but side lengths twice as large.

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A Reliable Strategy for Translations

1. Determine horizontal movement.
2. Determine vertical movement.
3. Write the rule.
4. Apply the same rule to every vertex.
5. Check that all points moved the same distance and direction.

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A Reliable Strategy for Reflections

1. Identify the line of reflection.
2. Choose the correct coordinate rule.
3. Transform each vertex.
4. Check that corresponding points are equally distant from the reflection line.
5. Confirm that the image has reversed orientation.

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A Reliable Strategy for Rotations

1. Identify the center of rotation.
2. Identify the angle.
3. Identify clockwise or counterclockwise direction.
4. Select the appropriate coordinate rule.
5. Apply it carefully to each point.
6. Check that each point remains the same distance from the center.

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A Reliable Strategy for Dilations

1. Identify the center of dilation.
2. Identify the scale factor.
3. Multiply coordinates appropriately.
4. Check whether the image should be larger or smaller.
5. Confirm that corresponding lengths are in the correct ratio.

For dilations centered at the origin:

(x, y) → (kx, ky)

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A Reliable Strategy for Multiple Transformations

1. Write the transformations in order.
2. Begin with the original coordinates.
3. Perform only the first transformation.
4. Record the new coordinates.
5. Use those coordinates for the next transformation.
6. Continue until all transformations are complete.
7. Check the final result.
8. Do not apply every transformation directly to the original coordinates.

A coordinate table can be helpful:

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Common Mistakes

Changing the wrong coordinate during a translation

Horizontal movement changes x. Vertical movement changes y.

Confusing reflection rules

Across x-axis:

(x, y) → (x, −y)

Across y-axis:

(x, y) → (−x, y)

Changing both signs for a 90° rotation

Changing both signs gives a 180° rotation, not a 90° rotation.

Confusing clockwise and counterclockwise rotations

90° counterclockwise:

(x, y) → (−y, x)

90° clockwise:

(x, y) → (y, −x)

Adding the scale factor during a dilation

A dilation multiplies coordinates.

For k = 3:

(x, y) → (3x, 3y)

not:

(x + 3, y + 3)

Assuming dilation preserves length

Dilation changes lengths by the scale factor.

Assuming area changes by k

Area changes by:

k²

Performing multiple transformations out of order

Transformation order can change the final result.

Applying the second transformation to the original point

The second transformation acts on the result of the first transformation.

Assuming every transformation preserves orientation

Reflections reverse orientation.

Forgetting the center of rotation or dilation

The familiar coordinate rules usually assume the origin. A different center requires additional reasoning.

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Key Terms

Transformation — operation that changes a figure's position, orientation, or size.

Preimage — original figure before a transformation.

Image — figure produced after a transformation.

Translation — transformation that slides every point the same distance and direction.

Translation vector — coordinate change describing the direction and distance of a translation.

Reflection — transformation that flips a figure across a line.

Line of reflection — line acting as the mirror in a reflection.

Rotation — transformation that turns a figure around a fixed point.

Center of rotation — fixed point around which a rotation occurs.

Clockwise — direction followed by the hands of a clock.

Counterclockwise — direction opposite the hands of a clock.

Dilation — transformation that changes size while preserving shape.

Scale factor — number determining how much a figure is enlarged or reduced.

Center of dilation — fixed point from which a dilation expands or contracts.

Rigid transformation — transformation that preserves lengths and angle measures.

Isometry — another term for a distance-preserving transformation.

Congruent figures — figures having the same size and shape.

Similar figures — figures having the same shape with proportional corresponding lengths.

Reflectional symmetry — symmetry in which reflection maps a figure onto itself.

Rotational symmetry — symmetry in which rotation maps a figure onto itself.

Composition of transformations — sequence of two or more transformations.

Inverse transformation — transformation that reverses another transformation.

Orientation — clockwise or counterclockwise order of vertices around a figure.

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Key Takeaways

• Transformations can change the position, orientation, or size of coordinate figures.
• Translations slide figures without changing their size, shape, or orientation.
• Reflections flip figures across a line and reverse orientation.
• Rotations turn figures around a center of rotation.
• Dilations change size according to a scale factor.
• Translation by (a, b) follows (x, y) → (x + a, y + b).
• Reflection across the x-axis follows (x, y) → (x, −y).
• Reflection across the y-axis follows (x, y) → (−x, y).
• Reflection across y = x follows (x, y) → (y, x).
• A 90° counterclockwise rotation about the origin follows (x, y) → (−y, x).
• A 90° clockwise rotation follows (x, y) → (y, −x).
• A 180° rotation follows (x, y) → (−x, −y).
• A dilation centered at the origin follows (x, y) → (kx, ky).
• Translations, reflections, and rotations are rigid transformations.
• Rigid transformations produce congruent figures.
• Dilations produce similar figures.
• Under dilation, lengths and perimeter change by k.
• Under dilation, area changes by k².
• Transformations preserve many geometric relationships, including angle measures, parallelism, and perpendicularity.
• Symmetry can be analyzed by determining whether a transformation maps a figure onto itself.
• Multiple transformations must generally be performed in the stated order.
• Inverse transformations can be used to recover the original coordinates.

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Check Your Understanding

1. Describe the difference between a translation, reflection, rotation, and dilation.
2. Point A(3, 5) is translated 4 units right and 7 units down. Find A′.
3. Triangle ABC has vertices A(1, 2), B(5, 2), and C(3, 6). Translate the triangle using (x, y) → (x − 3, y + 4).
4. Point P(6, −3) is reflected across the x-axis. Find P′.
5. Point Q(−4, 7) is reflected across the y-axis. Find Q′.
6. Reflect A(3, −5) across the line y = x.
7. Rotate B(4, 7) 90° counterclockwise about the origin.
8. Rotate C(−3, 6) 90° clockwise about the origin.
9. Rotate D(5, −2) 180° about the origin.
10. Triangle ABC has vertices A(1, 1), B(4, 1), and C(2, 3). Dilate the triangle about the origin using k = 3. Determine the new coordinates.
11. A polygon has perimeter 24 cm and area 30 cm². It is dilated by a scale factor of 2. Determine its new perimeter and area.
12. A square is reflected across the y-axis. Explain which geometric properties remain unchanged and which property related to orientation changes.
13. Point A(2, 1) is first translated 3 units right and then reflected across the y-axis. Find the final coordinates. Then reverse the order of the transformations and compare the results.
14. Triangle ABC has vertices A(1, 2), B(4, 2), and C(2, 5). First rotate it 90° counterclockwise about the origin and then dilate the resulting image by k = 2. Find the final coordinates.
15. A polygon has vertices A(2, 1), B(6, 1), C(6, 4), and D(2, 4). Determine whether the figure has reflectional symmetry about the horizontal line y = 2.5, and explain how coordinate relationships can be used to verify your conclusion.