Solved Exercises on Sequences of Rigid Transformations in High School Geometry

Master sequences of rigid transformations: compositions, order of operations, properties, and applications through these 5 detailed exercises.

Rules and methods, laws,...
\((T \circ R)(P) = T(R(P))\)
Composition of Transformations
\(r_x \circ r_y = R_{O,180°}\)
Composition of Reflections
Sequence
Rigid Composition
Preserves distance, angles
Order Matters
Non-commutative
AB ≠ BA in general
Result
Rigid Transformation
Overall preserves properties
Key definitions:

Sequence of Transformations: A series of transformations applied one after another. The composition T₁ ∘ T₂ means apply T₂ first, then apply T₁ to the result.

Composition Notation: (T₁ ∘ T₂)(P) = T₁(T₂(P)). Read from right to left: apply T₂ first, then T₁.

Pre-image: The original figure before the sequence of transformations.

Final Image: The figure after all transformations have been applied.

Sequence Methods:
  1. Identify the sequence: List transformations in order
  2. Apply transformations sequentially: Apply each transformation to the result of the previous one
  3. Track coordinates: Keep track of point coordinates through each step
  4. Verify the result: Check that the final transformation is rigid
  5. Confirm properties: Verify the composition preserves distances and angles
Tip 1: Order matters! Apply transformations right to left!
Tip 2: Each step in the sequence must be rigid.
Tip 3: The overall composition is also rigid.
Tip 4: Some sequences simplify to single transformations.
Solution: Exercises 1 to 3
1 Translation Then Reflection
Exercise 1
Apply translation T⟨2, 3⟩ followed by reflection across the x-axis to point P(1, -1). Find the final coordinates.
Definition:

Composition of Transformations: Applying transformations in sequence where (T₁ ∘ T₂)(P) means apply T₂ first, then T₁. The order is crucial for the result.

Composition Method:
  1. Apply the first transformation (rightmost): translation T⟨2, 3⟩
  2. Apply the second transformation: reflection across x-axis
  3. Track the point through each step
Original Point
P(1, -1)
First: Translation
T⟨2, 3⟩
Second: Reflection
Across x-axis
Final Point
P'(3, 4)
Step 1: Apply translation T⟨2, 3⟩ to P(1, -1)

Rule: (x, y) → (x+2, y+3)

P(1, -1) → P₁(1+2, -1+3) = P₁(3, 2)

Step 2: Apply reflection across x-axis to P₁(3, 2)

Rule: (x, y) → (x, -y)

P₁(3, 2) → P'(3, -2)

Step 3: Verify the result

Distance from origin to P: √[1² + (-1)²] = √2

Distance from origin to P': √[3² + (-2)²] = √13

Since individual distances changed, verify that the transformation preserves distances between points

Step 4: Confirm composition is rigid

Both translation and reflection are rigid transformations

Composition of rigid transformations is rigid ✓

Final coordinates: P'(3, -2)
Composition (r_x ∘ T⟨2,3⟩)(P)
Final answer:

The final coordinates after applying translation T⟨2, 3⟩ followed by reflection across the x-axis to point P(1, -1) are P'(3, -2). The composition preserves all geometric properties.

Applied rules:

• Order of Operations: Apply transformations right to left

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

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

2 Rotation Then Translation
Exercise 2
Apply 90° counterclockwise rotation about origin followed by translation T⟨-1, 2⟩ to point Q(3, 1). Find the final coordinates.
Definition:

Rotation Then Translation: First apply the rotation (x, y) → (-y, x) for 90° CCW, then apply the translation by adding the translation vector.

Original Point
Q(3, 1)
First: Rotation
90° CCW about origin
Second: Translation
T⟨-1, 2⟩
Final Point
Q'(-3, 5)
Step 1: Apply 90° CCW rotation about origin to Q(3, 1)

Rule: (x, y) → (-y, x)

Q(3, 1) → Q₁(-1, 3)

Step 2: Apply translation T⟨-1, 2⟩ to Q₁(-1, 3)

Rule: (x, y) → (x-1, y+2)

Q₁(-1, 3) → Q'(-1-1, 3+2) = Q'(-2, 5)

Step 3: Verify distance preservation

Distance from origin to Q: √[3² + 1²] = √10

Distance from origin to Q': √[(-2)² + 5²] = √29

Since distances to origin changed, verify with a distance between points

Step 4: Confirm the sequence is rigid

Both rotation and translation are rigid transformations

Composition of rigid transformations is rigid ✓

Final coordinates: Q'(-2, 5)
Composition (T⟨-1,2⟩ ∘ R_{O,90°})(Q)
Final answer:

The final coordinates after applying 90° CCW rotation about origin followed by translation T⟨-1, 2⟩ to point Q(3, 1) are Q'(-2, 5). The composition preserves all geometric properties.

Applied rules:

• Rotation Rule: 90° CCW: (x,y) → (-y, x)

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

• Composition Property: Rigid ∘ Rigid = Rigid

3 Two Reflections
Exercise 3
Apply reflection across y-axis followed by reflection across x-axis to point R(2, -3). Find the final coordinates and identify the equivalent single transformation.
Definition:

Composition of Two Reflections: Two reflections across perpendicular lines is equivalent to a rotation of 180° about their intersection point.

Original Point
R(2, -3)
First: Reflection
Across y-axis
Second: Reflection
Across x-axis
Final Point
R'(-2, 3)
Step 1: Apply reflection across y-axis to R(2, -3)

Rule: (x, y) → (-x, y)

R(2, -3) → R₁(-2, -3)

Step 2: Apply reflection across x-axis to R₁(-2, -3)

Rule: (x, y) → (x, -y)

R₁(-2, -3) → R'(-2, -(-3)) = R'(-2, 3)

Step 3: Identify the equivalent transformation

R(2, -3) → R'(-2, 3)

This is the same as (x, y) → (-x, -y)

Which is a 180° rotation about origin!

Step 4: Verify the equivalence

For 180° rotation: (x, y) → (-x, -y)

R(2, -3) → (-2, 3) ✓

This matches our result ✓

Step 5: Confirm the general property

Two reflections across perpendicular lines (x-axis and y-axis) = 180° rotation about their intersection (origin)

Final coordinates: R'(-2, 3)
Equivalent to: 180° rotation about origin
Final answer:

The final coordinates after applying reflection across y-axis followed by reflection across x-axis to point R(2, -3) are R'(-2, 3). This sequence is equivalent to a 180° rotation about the origin.

Applied rules:

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

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

• Composition Identity: r_x ∘ r_y = R_{O,180°}

Solution: Exercises 4 to 5
4 Three Transformations
Exercise 4
Apply translation T⟨1, -2⟩, then 90° CCW rotation about origin, then reflection across y-axis to point S(-1, 2). Find the final coordinates.
Definition:

Three-Step Composition: For (T₁ ∘ T₂ ∘ T₃)(P), apply T₃ first, then T₂, then T₁. Track the point through each transformation sequentially.

Original Point
S(-1, 2)
First: Translation
T⟨1, -2⟩
Second: Rotation
90° CCW about O
Third: Reflection
Across y-axis
Final Point
S'(4, 0)
Step 1: Apply translation T⟨1, -2⟩ to S(-1, 2)

Rule: (x, y) → (x+1, y-2)

S(-1, 2) → S₁(-1+1, 2-2) = S₁(0, 0)

Step 2: Apply 90° CCW rotation about origin to S₁(0, 0)

Rule: (x, y) → (-y, x)

S₁(0, 0) → S₂(-0, 0) = S₂(0, 0)

Step 3: Apply reflection across y-axis to S₂(0, 0)

Rule: (x, y) → (-x, y)

S₂(0, 0) → S'(-0, 0) = S'(0, 0)

Step 4: Verify with different starting point

Let's try with a point that's not the origin to see the effect

For T(-1, 2) to (0, 0) to (0, 0) to (0, 0)

Wait, let me recalculate: S(-1, 2)

Step 4 (corrected): Reapply transformations carefully

S(-1, 2) → translation → S₁(0, 0) → rotation → S₂(0, 0) → reflection → S'(0, 0)

Since the first transformation moved the point to the origin, and the origin is fixed by both rotation and reflection, the final result is S'(0, 0)

Step 5: Verify the composition is rigid

All three transformations (translation, rotation, reflection) are rigid

Composition of rigid transformations is rigid ✓

Final coordinates: S'(0, 0)
All transformations applied in sequence
Final answer:

The final coordinates after applying translation T⟨1, -2⟩, then 90° CCW rotation about origin, then reflection across y-axis to point S(-1, 2) are S'(0, 0).

Applied rules:

• Sequential Application: Apply transformations in order specified

• Fixed Points: Origin is fixed by rotation and reflection

• Rigidity: Composition of rigid transformations is rigid

5 Distance Preservation in Composition
Exercise 5
Points A(1, 0) and B(0, 1) undergo the sequence: 90° CCW rotation about origin, then translation T⟨2, -1⟩. Verify that distance AB equals distance A'B'.
Definition:

Rigid Composition Property: Since both rotation and translation are rigid transformations, their composition is also rigid, meaning it preserves distances between points.

Original Points
A(1,0), B(0,1)
First: Rotation
90° CCW about O
Second: Translation
T⟨2, -1⟩
Final Points
A'(2, 0), B'(1, 1)
Step 1: Apply transformations to point A(1, 0)

Rotation: A(1, 0) → A₁(0, 1) [using (x,y) → (-y, x)]

Translation: A₁(0, 1) → A'(0+2, 1-1) = A'(2, 0)

Step 2: Apply transformations to point B(0, 1)

Rotation: B(0, 1) → B₁(-1, 0) [using (x,y) → (-y, x)]

Translation: B₁(-1, 0) → B'(-1+2, 0-1) = B'(1, -1)

Step 3: Calculate distance AB using distance formula

AB = √[(0-1)² + (1-0)²] = √[1 + 1] = √2

Step 4: Calculate distance A'B' using distance formula

A'B' = √[(1-2)² + (-1-0)²] = √[1 + 1] = √2

Step 5: Compare distances

Distance AB = √2 units

Distance A'B' = √2 units

AB = A'B' ✓

Step 6: Confirm composition property

Since both rotation and translation are rigid transformations

Their composition is also rigid, preserving all distances ✓

AB = A'B' = √2 units
Composition preserves distances
Final answer:

The distance between A and B is √2 units, and the distance between A' and B' is also √2 units. This verifies that the composition of rigid transformations preserves distances.

Applied rules:

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

• Composition Property: Rigid ∘ Rigid = Rigid

• Verification: Compare distances before and after transformation

Key Concepts: Sequences of Rigid Transformations

Composition of Rigid Transformations

📊
Composition Rules

For composition (T₁ ∘ T₂)(P) = T₁(T₂(P)):

Apply transformations right to left
Rigid ∘ Rigid = Rigid
Order usually matters (non-commutative)
Common Compositions

Important equivalences:

r_x ∘ r_y = R_{O,180°}
R_θ ∘ R_φ = R_{θ+φ}
T_v ∘ T_w = T_{v+w}
Properties Preserved

Composition of rigid transformations preserves:

• Distance between points

• Angle measures

• Parallelism

• Congruence

• Orientation (if even number of reflections)

Composition Process
1
List Transformations
2
Apply Right-to-Left
3
Track Through Each Step
4
Verify Rigidity
5
Check Properties
Transformation Orders
(T ∘ R)(P) ≠ (R ∘ T)(P)
r_x ∘ r_y ≠ r_y ∘ r_x
R_θ ∘ T ≠ T ∘ R_θ
R_θ ∘ R_φ ≠ R_φ ∘ R_θ
(T ∘ R) ∘ S = T ∘ (R ∘ S)
Associative Property
Rigid Composition
Preserves Distance
Preserves Angles
Preserves Shape
Preserves Size
Creates Congruence
Order Matters!

Questions & Answers

Question: Why does the order of transformations matter? Isn't multiplication commutative?

Answer: Function composition (which is what transformations are) is generally not commutative, unlike multiplication of numbers. Consider this example:

Start with point (1, 0).

First translation T⟨1,0⟩ then reflection across y-axis: (1,0) → (2,0) → (-2,0)

First reflection across y-axis then translation T⟨1,0⟩: (1,0) → (-1,0) → (0,0)

These give different results! The order matters because each transformation changes the position of the point, so the next transformation operates on a different input.

Question: Can a sequence of transformations ever simplify to a single transformation?

Answer: Yes, many sequences of transformations simplify to single transformations! For example:

  • Two reflections across parallel lines = single translation
  • Two reflections across perpendicular lines = 180° rotation
  • Two rotations about the same point = single rotation
  • Two translations = single translation

This is why understanding compositions is valuable - you can often simplify complex sequences into simpler equivalent transformations.

Question: Is the composition of two rigid transformations always rigid? What about three or more?

Answer: Yes! The composition of any number of rigid transformations is always rigid. This is a fundamental property of rigid transformations.

Since rigid transformations preserve distances and angles, and composition preserves these properties:

  • Translation + Rotation = Rigid
  • Rotation + Reflection = Rigid
  • Reflection + Translation + Rotation = Rigid
  • Any combination of rigid transformations = Rigid

This is why sequences of rigid transformations are so useful in geometry - they maintain all the important properties of the original figure.

Question: I'm confused about the notation (T₁ ∘ T₂)(P). What does this mean exactly?

Answer: The notation (T₁ ∘ T₂)(P) means "T₁ composed with T₂ applied to P", and it's read from right to left:

(T₁ ∘ T₂)(P) = T₁(T₂(P))

This means:

  1. First apply T₂ to P
  2. Then apply T₁ to the result of step 1

Think of it like function composition: if f(x) = x² and g(x) = x+1, then (f ∘ g)(x) = f(g(x)) = f(x+1) = (x+1)².

So for transformations, you apply them in the order shown from right to left.