Solved Exercises on Introduction to Rigid Transformations in High School Geometry

Master rigid transformations: translations, rotations, reflections, and their preservation properties through these 5 detailed exercises.

Rules and methods, laws,...
\(T_{(a,b)}(x,y) = (x+a, y+b)\)
Translation Rule
\(R_{O,θ}(x,y) = (x', y')\)
Rotation Rule
Rigid Transformation
Preserves Distance
Lengths remain unchanged
Rigid Transformation
Preserves Angles
Angle measures stay same
Rigid Transformation
Preserves Orientation
Shape's handedness preserved
Key definitions:

Rigid Transformation: A transformation that preserves distances and angles between points. Also called an isometry.

Pre-image: The original figure before transformation.

Image: The figure after transformation.

Mapping: The function that takes each point of the pre-image to its corresponding point in the image.

Transformation Methods:
  1. Translation: Slide figure in a specific direction and distance
  2. Rotation: Turn figure around a fixed point by a specific angle
  3. Reflection: Flip figure across a line of reflection
  4. Verify rigidity: Check that distances and angles are preserved
  5. Notation: Use prime notation (A' for transformed point A)
Tip 1: Rigid transformations preserve all geometric properties!
Tip 2: Congruent figures are related by rigid transformations.
Tip 3: Order matters in compositions of transformations.
Tip 4: Check your work by verifying distance preservation.
Solution: Exercises 1 to 3
1 Translation Transformation
Exercise 1
Triangle ABC has vertices A(2, 3), B(4, 1), C(1, -1). Apply translation T(3, -2). Find coordinates of A'B'C'.
Definition:

Translation: A rigid transformation that moves every point of a figure the same distance in the same direction. The rule is T(a,b)(x,y) = (x+a, y+b).

Translation Method:
  1. Add the horizontal shift to each x-coordinate
  2. Add the vertical shift to each y-coordinate
  3. Verify that distances are preserved
Pre-image
A(2,3), B(4,1), C(1,-1)
Translation Vector
(3, -2)
Image
A'(5,1), B'(7,-1), C'(4,-3)
Step 1: Apply translation rule to each vertex

For T(3, -2): add 3 to x-coordinate, subtract 2 from y-coordinate

A(2, 3) → A'(2+3, 3-2) = A'(5, 1)

B(4, 1) → B'(4+3, 1-2) = B'(7, -1)

C(1, -1) → C'(1+3, -1-2) = C'(4, -3)

Step 2: Verify distance preservation

Calculate distance AB: √[(4-2)² + (1-3)²] = √[4 + 4] = √8

Calculate distance A'B': √[(7-5)² + (-1-1)²] = √[4 + 4] = √8

Distances are preserved ✓

Step 3: Verify angle preservation

Original triangle and translated triangle are congruent

All angles remain the same ✓

A'(5, 1), B'(7, -1), C'(4, -3)
Triangle A'B'C' is congruent to triangle ABC
Final answer:

The coordinates of the translated triangle are A'(5, 1), B'(7, -1), and C'(4, -3). The transformation is rigid as distances and angles are preserved.

Applied rules:

• Translation Rule: T(a,b)(x,y) = (x+a, y+b)

• Distance Formula: d = √[(x₂-x₁)² + (y₂-y₁)²]

• Rigidity Verification: Distances remain unchanged

2 Rotation Transformation
Exercise 2
Point P(3, 4) is rotated 90° counterclockwise about the origin. Find the coordinates of P'.
Definition:

Rotation: A rigid transformation that turns a figure around a fixed point (center of rotation) by a specific angle.

Original Point
P(3, 4)
Rotation
90° CCW about origin
Rotated Point
P'(-4, 3)
Step 1: Apply 90° CCW rotation rule

For 90° counterclockwise rotation about origin: (x, y) → (-y, x)

P(3, 4) → P'(-4, 3)

Step 2: Verify distance preservation

Distance from origin to P: √[3² + 4²] = √[9 + 16] = √25 = 5

Distance from origin to P': √[(-4)² + 3²] = √[16 + 9] = √25 = 5

Distance preserved ✓

Step 3: Understand the rotation pattern

Common rotation rules about origin:

90° CCW: (x, y) → (-y, x)

180°: (x, y) → (-x, -y)

270° CCW: (x, y) → (y, -x)

Step 4: Verify angle preservation

Any angle formed with the center of rotation is preserved

Rotation is a rigid transformation ✓

P'(-4, 3)
Distance from origin preserved: 5 units
Final answer:

The coordinates of the rotated point are P'(-4, 3). The rotation is rigid as the distance from the center of rotation is preserved.

Applied rules:

• 90° CCW Rotation: (x, y) → (-y, x)

• Distance Preservation: Distance from center remains same

• Rigidity: Rotations preserve all distances and angles

3 Reflection Transformation
Exercise 3
Point Q(5, -2) is reflected across the y-axis. Find the coordinates of Q'.
Definition:

Reflection: A rigid transformation that flips a figure across a line of reflection, creating a mirror image.

Original Point
Q(5, -2)
Reflection Line
y-axis (x = 0)
Reflected Point
Q'(-5, -2)
Step 1: Apply reflection rule across y-axis

For reflection across y-axis: (x, y) → (-x, y)

Q(5, -2) → Q'(-5, -2)

Step 2: Verify distance preservation

Distance from Q to y-axis: |5 - 0| = 5

Distance from Q' to y-axis: |-5 - 0| = 5

Distances to line of reflection are equal ✓

Step 3: Verify perpendicularity

The line segment QQ' is perpendicular to the y-axis

Midpoint of QQ': ((5-5)/2, (-2-2)/2) = (0, -2), which lies on y-axis ✓

Step 4: Understand reflection patterns

Common reflection rules:

Across y-axis: (x, y) → (-x, y)

Across x-axis: (x, y) → (x, -y)

Across origin: (x, y) → (-x, -y)

Step 5: Verify rigidity

Reflection preserves all distances and angles

It's a rigid transformation ✓

Q'(-5, -2)
Distance to y-axis preserved: 5 units
Final answer:

The coordinates of the reflected point are Q'(-5, -2). The reflection is rigid as distances to the line of reflection are preserved.

Applied rules:

• Reflection Across Y-axis: (x, y) → (-x, y)

• Distance to Mirror: Preserved on both sides

• Perpendicular Bisector: Line connecting point and image is perpendicular to mirror line

Solution: Exercises 4 to 5
4 Composition of Transformations
Exercise 4
Apply translation T(-2, 3) followed by reflection across x-axis to point R(4, 5). Find the final coordinates.
Definition:

Composition of Transformations: Applying multiple transformations in sequence. Each transformation is rigid, so the composition is also rigid.

Initial Point
R(4, 5)
First: Translation
T(-2, 3)
Second: Reflection
Across x-axis
Step 1: Apply translation T(-2, 3) to R(4, 5)

T(-2, 3)(4, 5) = (4-2, 5+3) = (2, 8)

After translation: R₁(2, 8)

Step 2: Apply reflection across x-axis to R₁(2, 8)

Reflection across x-axis: (x, y) → (x, -y)

(2, 8) → (2, -8)

Final point: R'(2, -8)

Step 3: Verify the composition is rigid

Translation preserves distances: ✓

Reflection preserves distances: ✓

Composition of rigid transformations is rigid: ✓

Step 4: Check distance preservation

Distance from origin to R: √[4² + 5²] = √[16 + 25] = √41

Distance from origin to R': √[2² + (-8)²] = √[4 + 64] = √68

Note: Individual distances to origin changed, but internal distances preserved

Step 5: Verify internal distance preservation

Compare distances between original and final positions

Since both transformations are rigid, the composition is rigid

Final coordinates: R'(2, -8)
Composition of rigid transformations is rigid
Final answer:

The final coordinates after the composition of transformations are R'(2, -8). The composition is rigid as both individual transformations are rigid.

Applied rules:

• Order of Operations: Apply transformations in specified sequence

• Composition Property: Composition of rigid transformations is rigid

• Step-by-Step Application: Apply each transformation to the result of the previous one

5 Identifying Rigid Transformations
Exercise 5
Given triangle DEF with vertices D(1, 1), E(4, 1), F(2, 4) and triangle D'E'F' with vertices D'(1, -1), E'(4, -1), F'(2, -4). Identify the transformation and verify it's rigid.
Definition:

Transformation Identification: Determining the specific rigid transformation that maps one figure to another by comparing corresponding points.

Pre-image
D(1,1), E(4,1), F(2,4)
Image
D'(1,-1), E'(4,-1), F'(2,-4)
Transformation
Reflection across x-axis
Step 1: Compare corresponding points

D(1, 1) → D'(1, -1)

E(4, 1) → E'(4, -1)

F(2, 4) → F'(2, -4)

Notice that x-coordinates remain the same, y-coordinates change sign

Step 2: Identify the transformation

Rule: (x, y) → (x, -y)

This is a reflection across the x-axis

Step 3: Verify distance preservation

Calculate DE in pre-image: √[(4-1)² + (1-1)²] = √[9 + 0] = 3

Calculate D'E' in image: √[(4-1)² + (-1-(-1))²] = √[9 + 0] = 3

Distance DE preserved ✓

Step 4: Verify all distances preserved

Calculate EF in pre-image: √[(2-4)² + (4-1)²] = √[4 + 9] = √13

Calculate E'F' in image: √[(2-4)² + (-4-(-1))²] = √[4 + 9] = √13

Distance EF preserved ✓

Step 5: Verify angle preservation

Since all sides are preserved, all angles are preserved

Triangles are congruent by SSS congruence

Step 6: Confirm transformation is rigid

Reflection across x-axis is a rigid transformation

All distances and angles preserved ✓

Transformation: Reflection across x-axis
Triangle D'E'F' ≅ Triangle DEF
All distances preserved
Final answer:

The transformation is a reflection across the x-axis. The transformation is rigid as all distances and angles are preserved, confirming that triangle D'E'F' is congruent to triangle DEF.

Applied rules:

• Transformation Identification: Compare patterns in coordinates

• Rigidity Verification: Check distance preservation

• Congruence: Rigid transformations produce congruent figures

Key Concepts: Introduction to Rigid Transformations

Rigid Transformations Fundamentals

📊
Translation Properties

Moves every point the same distance in the same direction:

T(a,b)(x,y) = (x+a, y+b)

Preserves: distances, angles, orientation, parallelism

Rotation Properties

Turns figure around a fixed point by specific angle:

R(center, angle)

Preserves: distances from center, angles, congruence

Reflection Properties

Flips figure across line of reflection:

Across line l

Preserves: distances to mirror line, angles, creates mirror image

Rigid Transformation Properties
✓
Distance Preserved
✓
Angle Preserved
✓
Parallelism Preserved
✓
Congruence Maintained
✓
Orientation Preserved
Transformation Types
↔️ Translation
🔄 Rotation
镜 Reflection
Key Properties
Distance: Unchanged
Angles: Unchanged
Area: Unchanged
Shape: Unchanged
Congruence: Maintained
Rigid = Preserves All!

Questions & Answers

Question: How do I know if a transformation is rigid? What exactly gets preserved?

Answer: A transformation is rigid (or an isometry) if it preserves all distances and angles. Specifically, rigid transformations preserve:

  • Distances: The distance between any two points remains the same
  • Angles: The measure of any angle remains the same
  • Areas: The area of any figure remains the same
  • Shapes: The overall shape remains the same (figures are congruent)
  • Parallelism: Parallel lines remain parallel

To verify a transformation is rigid, you can check that the distance between any two points in the pre-image equals the distance between their corresponding points in the image.

Question: What's the difference between a rotation of 90° counterclockwise and 270° clockwise? Are they the same transformation?

Answer: Yes, a rotation of 90° counterclockwise and 270° clockwise are equivalent transformations. They both result in the same final position.

For example, starting with point (x, y):

  • 90° CCW: (x, y) → (-y, x)
  • 270° CW: (x, y) → (-y, x)

This is because 270° clockwise = 360° - 270° = 90° counterclockwise.

In general, rotating θ degrees clockwise is equivalent to rotating (360° - θ) degrees counterclockwise.

Question: Does reflection change the orientation of a figure? I noticed that letters can be "flipped."

Answer: Yes, reflection is the only rigid transformation that changes orientation (also called "handedness").

Consider the letter "R":

  • In the original position, the "arm" extends to the upper right
  • After reflection, the "arm" extends to the upper left
  • The letter appears as a mirror image

Translations and rotations preserve orientation, but reflections reverse it. However, all rigid transformations preserve distances and angles, so the figure remains congruent to its original.

This is why reflection is sometimes called an "improper" isometry while translation and rotation are "proper" isometries.

Question: When composing transformations, does the order matter? Could I get different results?

Answer: Yes, the order of transformations usually matters! Composition of transformations is generally not commutative.

For example, consider point (1, 1):

Translation T(1, 0) then reflection across y-axis:

  • (1, 1) → (2, 1) → (-2, 1)

Reflection across y-axis then translation T(1, 0):

  • (1, 1) → (-1, 1) → (0, 1)

These give different results! The order matters unless the transformations commute (which happens in special cases).

Always perform transformations in the order specified.