Geometric Transformations

A geometric transformation changes the position, orientation, or size of a figure.

Two important transformations are:

• Reflection – flips a figure across a line.
• Rotation – turns a figure around a fixed point.

Like translations, reflections and rotations are rigid transformations. This means they preserve:

• Side lengths.
• Angle measures.
• Size.
• Shape.

The original figure and its transformed image are therefore congruent.

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Reflections

A reflection is a transformation that flips a figure across a line.

The result looks like a mirror image of the original figure.

The line across which the figure is flipped is called the:

line of reflection

Imagine placing a mirror along this line. The reflected figure would appear on the opposite side.

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The Line of Reflection

Every point and its reflected image are the same perpendicular distance from the line of reflection.

For example, if point A is:

3 units to the right

of a vertical line of reflection, then A′ will be:

3 units to the left

of that line.

The line of reflection lies exactly halfway between corresponding points.

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Reflection Across the x-Axis

One of the most common coordinate transformations is reflection across the x-axis.

Suppose:

A(3, 4)

is reflected across the x-axis.

The x-coordinate stays the same.

The y-coordinate changes sign.

Therefore:

A(3, 4) → A′(3, −4)

The rule is:

(x, y) → (x, −y)

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Why the y-Coordinate Changes

The x-axis is horizontal.

A reflection across it changes the point's vertical position.

For example:

(2, 5) → (2, −5)

The original point is 5 units above the x-axis.

The reflected point must be 5 units below the x-axis.

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Another x-Axis Example

Reflect:

B(−4, −3)

across the x-axis.

Keep x:

−4

Change the sign of y:

−3 → 3

Therefore:

B′(−4, 3)

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Reflection Across the y-Axis

When reflecting across the y-axis, the y-coordinate stays the same.

The x-coordinate changes sign.

For example:

C(4, 2) → C′(−4, 2)

The rule is:

(x, y) → (−x, y)

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Why the x-Coordinate Changes

The y-axis is vertical.

A reflection across it changes the point's horizontal position.

For example:

(6, 3)

is 6 units to the right of the y-axis.

Its reflection must be 6 units to the left.

Therefore:

(6, 3) → (−6, 3)

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Reflection Across the Line y = x

Another important line of reflection is:

y = x

When reflecting across this line, the x- and y-coordinates exchange positions.

The rule is:

(x, y) → (y, x)

For example:

(2, 5) → (5, 2)

and:

(−3, 4) → (4, −3)

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Reflecting an Entire Figure

Suppose triangle ABC has vertices:

A(1, 2)

B(4, 2)

C(2, 5)

Reflect the triangle across the y-axis.

Use:

(x, y) → (−x, y)

Therefore:

A(1, 2) → A′(−1, 2)

B(4, 2) → B′(−4, 2)

C(2, 5) → C′(−2, 5)

Every vertex follows the same reflection rule.

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What Reflections Preserve

A reflection preserves:

• Side lengths.
• Angle measures.
• Parallel lines.
• Perpendicular lines.
• Area.
• Perimeter.
• Overall size and shape.

However, a reflection reverses orientation.

This is an important difference between reflections and translations.

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Orientation After Reflection

Imagine a triangle whose vertices go around the figure in this order:

A → B → C

After reflection, its mirror image has the opposite orientation.

This is similar to looking at writing in a mirror.

The shape has not changed size, but it has been:

flipped

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Lines of Reflection in Shapes

A line of reflection can also refer to a line of symmetry within a figure.

If a shape can be folded along a line so that the two halves match exactly, that line is a:

line of symmetry

or:

line of reflection

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Lines of Reflection in a Square

A square has:

4 lines of reflection

These are:

• One vertical line through the center.
• One horizontal line through the center.
• Two diagonals.

Each line divides the square into two matching halves.

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Lines of Reflection in a Rectangle

A non-square rectangle has:

2 lines of reflection

They pass:

• Vertically through the center.
• Horizontally through the center.

Its diagonals are not lines of reflection unless the rectangle is a square.

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Lines of Reflection in Triangles

An equilateral triangle has:

3 lines of reflection

An isosceles triangle has:

1 line of reflection

A scalene triangle has:

0 lines of reflection

This provides another way to compare triangle types.

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Lines of Reflection in Circles

A circle has:

infinitely many lines of reflection

Any line passing through its center divides the circle into two matching halves.

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Rotations

A rotation is a transformation that turns a figure around a fixed point.

The fixed point is called the:

center of rotation

A rotation must specify:

• Center of rotation.
• Angle of rotation.
• Direction of rotation.

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Clockwise and Counterclockwise

A rotation can occur in two directions.

Clockwise means turning in the same direction as the hands of a clock.

Counterclockwise means turning in the opposite direction.

For example:

90° clockwise

and:

90° counterclockwise

produce different positions.

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Common Rotation Angles

Common rotation angles include:

90°

180°

270°

360°

A 360° rotation is one complete turn.

After a full 360° rotation, a figure returns to its original position.

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Quarter, Half, and Full Turns

It can be helpful to connect angles with turns.

90° = ¼ turn

180° = ½ turn

270° = ¾ turn

360° = 1 complete turn

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Rotations About the Origin

On a coordinate plane, rotations are often performed around:

the origin (0, 0)

There are useful coordinate rules for these rotations.

For a 90° counterclockwise rotation:

(x, y) → (−y, x)

For a 180° rotation:

(x, y) → (−x, −y)

For a 270° counterclockwise rotation:

(x, y) → (y, −x)

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90° Counterclockwise Rotation

Consider:

A(3, 2)

Rotate it 90° counterclockwise around the origin.

Use:

(x, y) → (−y, x)

Therefore:

(3, 2) → (−2, 3)

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Another 90° Counterclockwise Example

Consider:

B(−4, 2)

Apply:

(x, y) → (−y, x)

Therefore:

B(−4, 2) → B′(−2, −4)

The point moves from Quadrant II to Quadrant III.

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90° Clockwise Rotation

For a 90° clockwise rotation around the origin:

(x, y) → (y, −x)

For example:

(3, 2) → (2, −3)

Notice that this is the same coordinate rule as a:

270° counterclockwise rotation

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180° Rotation

For a 180° rotation around the origin:

(x, y) → (−x, −y)

Both coordinates change signs.

For example:

(4, 3) → (−4, −3)

Another example:

(−5, 2) → (5, −2)

The direction does not matter for a 180° rotation.

A 180° clockwise rotation and a 180° counterclockwise rotation produce the same result.

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270° Counterclockwise Rotation

For a 270° counterclockwise rotation:

(x, y) → (y, −x)

For example:

(2, 5) → (5, −2)

A 270° counterclockwise rotation is equivalent to:

90° clockwise

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Rotation Rules About the Origin

These rules apply when the center of rotation is the origin.

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Understanding Rather Than Memorizing

Rotation rules can be memorized, but understanding the movement is more useful.

Consider:

A(4, 1)

The point lies mostly to the right and slightly above the origin.

After a 90° counterclockwise turn, it should lie:

• Mostly above the origin.
• Slightly to the left.

The result:

A′(−1, 4)

makes sense.

A quick sketch can help check whether a coordinate rule has been used correctly.

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Rotating an Entire Figure

Suppose triangle ABC has:

A(1, 1)

B(4, 1)

C(2, 3)

Rotate it 90° counterclockwise around the origin.

Use:

(x, y) → (−y, x)

For A:

A(1, 1) → A′(−1, 1)

For B:

B(4, 1) → B′(−1, 4)

For C:

C(2, 3) → C′(−3, 2)

The entire triangle has turned around the origin.

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Rotating About a Point Other Than the Origin

A figure can rotate around any point.

Suppose point A is rotated around point P.

Point P remains fixed while A travels along part of a circular path around it.

The distance between A and P remains constant.

This gives us an important rotation property:

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

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Finding the Center of Rotation

When examining a diagram, the center of rotation is the fixed point around which the figure turns.

To identify it, consider corresponding points such as:

A and A′

The center must be equally distant from both.

The same must be true for:

B and B′

and other corresponding points.

For basic problems, the center is often given directly.

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Rotations Preserve Size and Shape

Rotations are rigid transformations.

They preserve:

• Side lengths.
• Angles.
• Area.
• Perimeter.
• Parallel relationships.
• Perpendicular relationships.

Therefore, the original figure and rotated image are:

congruent

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Rotations Preserve Orientation

Unlike a reflection, a rotation does not create a mirror image.

A rotation turns the figure but preserves the order of its vertices.

This means its orientation is preserved.

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Comparing Reflections and Rotations

Reflections and rotations are both rigid transformations.

They both preserve:

• Size.
• Shape.
• Side lengths.
• Angle measures.
• Area.
• Perimeter.

However, they move figures differently.

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Reflection Versus Rotation Example

Suppose an arrow points:

right

After reflection across a vertical line, it may point:

left

It has been flipped.

After a 90° counterclockwise rotation, it would point:

up

It has been turned.

The two transformations therefore produce different results even though both preserve the arrow's size and shape.

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Comparing Translation, Reflection, and Rotation

The three rigid transformations can be remembered as:

Translation → slide

Reflection → flip

Rotation → turn

All three preserve:

• Size.
• Shape.
• Lengths.
• Angles.

But they change position and sometimes orientation.

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Reflection Rules

Some useful coordinate rules are:

Across the x-axis

(x, y) → (x, −y)

Across the y-axis

(x, y) → (−x, y)

Across y = x

(x, y) → (y, x)

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Rotation Rules

For rotations around the origin:

90° counterclockwise

(x, y) → (−y, x)

180°

(x, y) → (−x, −y)

270° counterclockwise

(x, y) → (y, −x)

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Worked Reflection Example

Rectangle ABCD has vertices:

A(1, 2)

B(5, 2)

C(5, 4)

D(1, 4)

Reflect the rectangle across the y-axis.

Use:

(x, y) → (−x, y)

Therefore:

A′(−1, 2)

B′(−5, 2)

C′(−5, 4)

D′(−1, 4)

The new rectangle has the same dimensions as the original but has been flipped across the y-axis.

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Worked Rotation Example

Triangle ABC has vertices:

A(2, 1)

B(5, 1)

C(3, 4)

Rotate the triangle:

180° about the origin

Use:

(x, y) → (−x, −y)

Therefore:

A′(−2, −1)

B′(−5, −1)

C′(−3, −4)

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Checking a Reflection

Suppose:

A(4, 3) → A′(−4, 3)

The y-coordinate remained unchanged.

The x-coordinate changed sign.

Therefore, this is consistent with a reflection across the:

y-axis

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Checking a Rotation

Suppose:

B(3, 5) → B′(−3, −5)

Both coordinates changed signs.

Therefore, this is consistent with:

a 180° rotation around the origin

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Symmetry and Reflection

A figure has reflectional symmetry if it can be reflected across a line and match itself exactly.

Examples include:

• Squares.
• Rectangles.
• Equilateral triangles.
• Isosceles triangles.
• Circles.

The number of lines of reflection depends on the shape.

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Rotational Symmetry

A figure has rotational symmetry if it can be rotated less than 360° around a point and still match its original position.

For example, a square matches itself after rotations of:

90°, 180°, 270°, and 360°

A non-square rectangle matches itself after:

180° and 360°

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Order of Rotational Symmetry

The order of rotational symmetry tells us how many times a figure matches itself during one complete 360° turn.

A square has rotational symmetry of order:

4

A non-square rectangle has order:

2

An equilateral triangle has order:

3

A general scalene triangle has order:

1

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Reflections and Rotations in the Real World

These transformations appear frequently in nature, design, art, and engineering.

Examples include:

• Reflections in mirrors.
• Butterfly wing patterns.
• Building designs.
• Decorative tiles.
• Wheels and gears.
• Fans and propellers.
• Logos.
• Flowers.
• Snowflake patterns.

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A Reliable Reflection Strategy

When reflecting a figure:

Step 1: Identify the line of reflection.

Step 2: Determine each vertex's perpendicular distance from the line.

Step 3: Place the reflected point the same distance on the opposite side.

Step 4: Repeat for every vertex.

Step 5: Connect the new points in the corresponding order.

Step 6: Check that size and shape have been preserved.

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A Reliable Rotation Strategy

When rotating a figure:

Step 1: Identify the center of rotation.

Step 2: Identify the angle.

Step 3: Identify whether the rotation is clockwise or counterclockwise.

Step 4: Rotate each vertex by the same angle.

Step 5: Plot the new coordinates.

Step 6: Connect the corresponding vertices.

Step 7: Check that every point remains the same distance from the center of rotation.

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Common Mistakes

Confusing Reflection and Rotation

Remember:

Reflection = flip

Rotation = turn

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Changing the Size of the Figure

Neither reflection nor rotation changes size.

The image must remain:

congruent

to the original.

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Using the Wrong Reflection Rule

Across the x-axis:

change the sign of y

Across the y-axis:

change the sign of x

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Changing Both Signs for Every Reflection

Changing both signs:

(x, y) → (−x, −y)

is not generally a reflection across the x- or y-axis.

It represents a:

180° rotation about the origin

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Forgetting the Direction of Rotation

A 90° clockwise rotation is different from a 90° counterclockwise rotation.

Always check the direction.

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Rotating Around the Wrong Point

A rotation must occur around a specific:

center of rotation

Rotating around the origin when another center is specified will produce the wrong result.

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Forgetting That 180° Is Special

A 180° clockwise rotation produces the same result as a 180° counterclockwise rotation.

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Assuming a Diagonal Is Always a Line of Symmetry

The diagonals of a square are lines of symmetry.

The diagonals of a non-square rectangle are not.

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Confusing Reflectional and Rotational Symmetry

Reflectional symmetry involves a:

flip across a line

Rotational symmetry involves a:

turn around a point

A shape may have one, both, or neither.

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Check Your Understanding

1. Define a reflection.

2. Define a rotation.

3. What is a line of reflection?

4. What is the center of rotation?

5. What three pieces of information describe a rotation?

6. Reflect (4, 3) across the x-axis.

7. Reflect (−5, 2) across the x-axis.

8. Reflect (6, −3) across the y-axis.

9. Reflect (−2, −7) across the y-axis.

10. Reflect (3, 5) across y = x.

11. Write the coordinate rule for reflection across the x-axis.

12. Write the coordinate rule for reflection across the y-axis.

13. Rotate (3, 2) 90° counterclockwise around the origin.

14. Rotate (4, 1) 90° clockwise around the origin.

15. Rotate (5, 3) 180° around the origin.

16. Rotate (−2, 4) 270° counterclockwise around the origin.

17. Write the rule for a 90° counterclockwise rotation around the origin.

18. Write the rule for a 180° rotation around the origin.

19. Explain why a reflection preserves size and shape.

20. Explain why a rotation preserves size and shape.

21. What happens to orientation during a reflection?

22. What happens to orientation during a rotation?

23. How many lines of reflection does a square have?

24. How many lines of reflection does a non-square rectangle have?

25. How many lines of reflection does an equilateral triangle have?

26. How many lines of reflection does a scalene triangle have?

27. What is the order of rotational symmetry of a square?

28. A point changes from (4, 6) to (−4, 6). Identify the likely transformation.

29. A point changes from (3, −5) to (−3, 5). Identify the rotation about the origin.

30. Explain one similarity and one difference between a reflection and a rotation.

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Key Terms

• Transformation – change in the position, orientation, or size of a geometric figure.
• Reflection – transformation that flips a figure across a line.
• Line of reflection – line across which a figure is reflected.
• Rotation – transformation that turns a figure around a fixed point.
• Center of rotation – fixed point around which a rotation occurs.
• Clockwise – direction in which the hands of a clock turn.
• Counterclockwise – direction opposite to clock hands.
• Preimage – original figure before a transformation.
• Image – resulting figure after a transformation.
• Congruent – having the same size and shape.
• Orientation – arrangement or direction of a figure and its vertices.
• Reflectional symmetry – symmetry created when a figure matches itself after reflection.
• Rotational symmetry – symmetry created when a figure matches itself after rotation.
• Order of rotational symmetry – number of times a figure matches itself during a complete 360° rotation.

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Key Takeaways

• A reflection flips a figure across a line.
• A rotation turns a figure around a fixed point.
• Reflections and rotations preserve size and shape.
• They also preserve side lengths and angle measures.
• A reflection reverses orientation.
• A rotation preserves orientation.
• Reflection across the x-axis:

(x, y) → (x, −y)

• Reflection across the y-axis:

(x, y) → (−x, y)

• Reflection across y = x:

(x, y) → (y, x)

• 90° counterclockwise rotation about the origin:

(x, y) → (−y, x)

• 180° rotation about the origin:

(x, y) → (−x, −y)

• 270° counterclockwise rotation about the origin:

(x, y) → (y, −x)

• Every point and its reflected image are equally distant from the line of reflection.
• Every point maintains its distance from the center during a rotation.
• Reflectional symmetry involves a line.
• Rotational symmetry involves a center and an angle.
• Reflections and rotations are examples of rigid transformations.

A useful way to distinguish the transformations is:

Translation = slide

Reflection = flip

Rotation = turn