Solved Exercises on Translations in High School Geometry

Master translations: vector notation, coordinate rules, geometric properties, and applications through these 5 detailed exercises.

Rules and methods, laws,...
\(T_{(a,b)}(x,y) = (x+a, y+b)\)
Translation Rule
\(\vec{v} = \langle a, b \rangle\)
Translation Vector
Translation
Rigid Transformation
Preserves distance, angles
Vector Notation
\(\langle a, b \rangle\)
Horizontal and vertical shifts
Coordinate Rule
Add vector components
(x, y) → (x+a, y+b)
Key definitions:

Translation: A rigid transformation that moves every point of a figure the same distance in the same direction. Every point (x, y) moves to (x+a, y+b).

Translation Vector: The directed line segment that specifies the direction and distance of the translation, written as ⟨a, b⟩.

Pre-image: The original figure before the transformation.

Image: The figure after the transformation, denoted with prime notation (A' for transformed point A).

Translation Methods:
  1. Identify the translation vector: Determine the horizontal (a) and vertical (b) components
  2. Apply the coordinate rule: Add vector components to each point (x, y) → (x+a, y+b)
  3. Verify the transformation: Check that distances and angles are preserved
  4. Graph the result: Plot the translated figure
  5. Confirm properties: Verify the transformation is rigid
Tip 1: Positive x-shift moves right, negative moves left!
Tip 2: Positive y-shift moves up, negative moves down!
Tip 3: Translations preserve all geometric properties.
Tip 4: Remember to apply the same translation to ALL points!
Solution: Exercises 1 to 3
1 Single Point Translation
Exercise 1
Point P(3, -2) is translated by vector ⟨-4, 5⟩. Find the coordinates of P'.
Definition:

Translation Vector: The ordered pair ⟨a, b⟩ indicates how far to move horizontally (a) and vertically (b).

Translation Method:
  1. Identify the translation vector: ⟨-4, 5⟩
  2. Apply the rule: (x, y) → (x+a, y+b)
  3. Calculate the new coordinates
Original Point
P(3, -2)
Translation Vector
⟨-4, 5⟩
Translated Point
P'(-1, 3)
Step 1: Identify the translation vector

The vector ⟨-4, 5⟩ means move 4 units left and 5 units up

Horizontal component: a = -4

Vertical component: b = 5

Step 2: Apply the translation rule

For translation T⟨a,b⟩, the rule is (x, y) → (x+a, y+b)

P(3, -2) → P'(3+(-4), -2+5)

Step 3: Calculate the new coordinates

P'(3-4, -2+5) = P'(-1, 3)

Step 4: Verify the movement

Started at (3, -2), moved 4 left to -1, moved 5 up to 3

Correct: P'(-1, 3)

P'(-1, 3)
Moved 4 units left and 5 units up
Final answer:

The coordinates of the translated point are P'(-1, 3). The point moved 4 units left and 5 units up from its original position.

Applied rules:

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

• Vector Components: Horizontal shift + Vertical shift

• Coordinate Addition: Add vector components to original coordinates

2 Triangle Translation
Exercise 2
Triangle ABC has vertices A(1, 2), B(4, 0), C(2, -3). Apply translation T⟨-2, 4⟩. Find coordinates of A'B'C'.
Definition:

Figure Translation: A transformation that moves every point of a figure by the same vector. All points move the same distance in the same direction.

Pre-image
A(1,2), B(4,0), C(2,-3)
Translation Vector
⟨-2, 4⟩
Image
A'(-1,6), B'(2,4), C'(0,1)
Step 1: Apply translation to vertex A

A(1, 2) → A'(1+(-2), 2+4) = A'(-1, 6)

Step 2: Apply translation to vertex B

B(4, 0) → B'(4+(-2), 0+4) = B'(2, 4)

Step 3: Apply translation to vertex C

C(2, -3) → C'(2+(-2), -3+4) = C'(0, 1)

Step 4: Verify all points moved the same distance

Each point moved 2 units left and 4 units up

All points moved according to the same vector ⟨-2, 4⟩

Step 5: Check that the triangle is preserved

Translation is a rigid transformation, so triangle A'B'C' is congruent to triangle ABC

A'(-1, 6), B'(2, 4), C'(0, 1)
Triangle A'B'C' ≅ Triangle ABC
Final answer:

The coordinates of the translated triangle are A'(-1, 6), B'(2, 4), and C'(0, 1). The translation preserved the shape and size of the triangle.

Applied rules:

• Uniform Translation: Apply same vector to all points

• Rigid Transformation: Preserves distances and angles

• Congruence: Original and translated figures are congruent

3 Finding Translation Vector
Exercise 3
Point D(5, -1) is translated to D'(2, 3). Find the translation vector and describe the movement.
Definition:

Translation Vector from Points: If point P(x₁, y₁) translates to P'(x₂, y₂), then the translation vector is ⟨x₂-x₁, y₂-y₁⟩.

Original Point
D(5, -1)
Translated Point
D'(2, 3)
Translation Vector
⟨-3, 4⟩
Step 1: Identify coordinates

Original point D: (x₁, y₁) = (5, -1)

Translated point D': (x₂, y₂) = (2, 3)

Step 2: Calculate horizontal component

Horizontal shift = x₂ - x₁ = 2 - 5 = -3

This means 3 units to the left

Step 3: Calculate vertical component

Vertical shift = y₂ - y₁ = 3 - (-1) = 3 + 1 = 4

This means 4 units up

Step 4: Write the translation vector

Translation vector = ⟨horizontal shift, vertical shift⟩ = ⟨-3, 4⟩

Step 5: Describe the movement

The point moved 3 units left and 4 units up

Verification: D(5, -1) + ⟨-3, 4⟩ = (5-3, -1+4) = (2, 3) = D' ✓

Translation vector: ⟨-3, 4⟩
Movement: 3 units left, 4 units up
Final answer:

The translation vector is ⟨-3, 4⟩, which means the point moved 3 units left and 4 units up from its original position.

Applied rules:

• Vector Calculation: ⟨x₂-x₁, y₂-y₁⟩

• Direction Interpretation: Negative = left/down, Positive = right/up

• Verification: Apply vector to original point to get image

Solution: Exercises 4 to 5
4 Composition of Translations
Exercise 4
Point Q(2, -1) is translated by ⟨3, 2⟩, then by ⟨-1, 4⟩. Find the final coordinates and the equivalent single translation.
Definition:

Composition of Translations: When two or more translations are applied sequentially, the result is equivalent to a single translation whose vector is the sum of the individual vectors.

Initial Point
Q(2, -1)
First Translation
⟨3, 2⟩
Second Translation
⟨-1, 4⟩
Final Point
Q'(4, 5)
Step 1: Apply first translation T⟨3, 2⟩ to Q(2, -1)

Q(2, -1) → Q₁(2+3, -1+2) = Q₁(5, 1)

Step 2: Apply second translation T⟨-1, 4⟩ to Q₁(5, 1)

Q₁(5, 1) → Q'(5+(-1), 1+4) = Q'(4, 5)

Step 3: Find equivalent single translation

Starting point: Q(2, -1)

Final point: Q'(4, 5)

Equivalent vector: ⟨4-2, 5-(-1)⟩ = ⟨2, 6⟩

Step 4: Verify by adding translation vectors

⟨3, 2⟩ + ⟨-1, 4⟩ = ⟨3+(-1), 2+4⟩ = ⟨2, 6⟩

This matches our calculated equivalent vector ✓

Step 5: Apply equivalent translation to verify

Q(2, -1) + ⟨2, 6⟩ = (2+2, -1+6) = (4, 5) = Q' ✓

Final coordinates: Q'(4, 5)
Equivalent translation: ⟨2, 6⟩
Final answer:

The final coordinates after both translations are Q'(4, 5). This is equivalent to a single translation of ⟨2, 6⟩, which is the sum of the individual translation vectors.

Applied rules:

• Vector Addition: Compositions of translations add the vectors

• Sequential Application: Apply transformations in order

• Equivalence: Multiple translations = single translation with summed vector

5 Translation and Distance Preservation
Exercise 5
Points A(1, 1) and B(4, 5) are translated by vector ⟨-2, 3⟩ to A' and B'. Verify that the distance between A and B equals the distance between A' and B'.
Definition:

Rigid Transformation Property: Translations are rigid transformations, meaning they preserve distances between points. The distance between any two points in the pre-image equals the distance between their images.

Original Points
A(1,1), B(4,5)
Translation Vector
⟨-2, 3⟩
Translated Points
A'(-1,4), B'(2,8)
Step 1: Find coordinates of A' and B'

A(1, 1) → A'(1+(-2), 1+3) = A'(-1, 4)

B(4, 5) → B'(4+(-2), 5+3) = B'(2, 8)

Step 2: Calculate distance AB using distance formula

d = √[(x₂-x₁)² + (y₂-y₁)²]

AB = √[(4-1)² + (5-1)²] = √[3² + 4²] = √[9 + 16] = √25 = 5

Step 3: Calculate distance A'B' using distance formula

A'B' = √[(2-(-1))² + (8-4)²] = √[3² + 4²] = √[9 + 16] = √25 = 5

Step 4: Compare distances

Distance AB = 5 units

Distance A'B' = 5 units

AB = A'B' ✓

Step 5: Verify the rigid transformation property

Since AB = A'B', the translation preserved the distance between points

This confirms that translation is a rigid transformation

AB = A'B' = 5 units
Translation preserves distances
Final answer:

The distance between A and B is 5 units, and the distance between A' and B' is also 5 units. This verifies that translation preserves distances, confirming it is a rigid transformation.

Applied rules:

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

• Rigidity Property: Translations preserve all distances

• Verification: Compare distances before and after transformation

Key Concepts: Translations

Translation Fundamentals

📊
Translation Rule

Basic translation rule:

T⟨a,b⟩(x,y) = (x+a, y+b)

Every point moves by the same vector ⟨a, b⟩

Vector Notation

Translation vectors specify movement:

⟨a, b⟩ where a = horizontal shift, b = vertical shift

Positive: right/up, Negative: left/down

Properties Preserved

Translations preserve all geometric properties:

• Distances between points

• Angle measures

• Parallelism

• Congruence

Translation Process
1
Identify Vector
2
Apply Rule
3
Calculate New Coordinates
4
Verify Properties
5
Graph Result
Vector Components
⟨+, +⟩ → NE
⟨-, +⟩ → NW
⟨+, -⟩ → SE
⟨-, -⟩ → SW
⟨0, +⟩ → N
⟨+, 0⟩ → E
Rigid Transformation
Preserves Distance
Preserves Angles
Preserves Shape
Preserves Size
Creates Congruence
Translation = Slide!

Questions & Answers

Question: How do I remember which direction corresponds to positive and negative values in the translation vector?

Answer: Remember the coordinate plane conventions:

  • Positive x-direction: Move right on the coordinate plane
  • Negative x-direction: Move left on the coordinate plane
  • Positive y-direction: Move up on the coordinate plane
  • Negative y-direction: Move down on the coordinate plane

For vector ⟨a, b⟩:

  • If a > 0: move right
  • If a < 0: move left
  • If b > 0: move up
  • If b < 0: move down

Mnemonic: "Right and Up are Positive" (RUP).

Question: If I translate a line, does the slope change? What about parallel lines?

Answer: No, translation does not change the slope of a line. Since translation is a rigid transformation:

  • Slope remains the same: The rise over run ratio stays constant
  • Parallel lines remain parallel: If two lines are parallel before translation, they remain parallel after translation
  • Perpendicular lines remain perpendicular: The 90-degree relationship is preserved

This is because translation preserves all geometric relationships and measurements. The only thing that changes is the position of the figure in the coordinate plane.

Question: Why are translations called "rigid" transformations? What makes them different from other transformations?

Answer: Translations are called "rigid" transformations because they preserve the "rigidity" or inflexibility of shapes. Specifically:

  • Distance preservation: The distance between any two points remains unchanged
  • Angle preservation: All angle measures remain the same
  • Shape preservation: The overall shape remains identical
  • Size preservation: Areas, perimeters, and all dimensions remain unchanged

This differs from non-rigid transformations like dilations, which change the size of figures while preserving shape (similarity). Rigid transformations maintain both shape and size, producing congruent figures.

Question: When working with translations, how do I handle fractional or decimal values in the translation vector?

Answer: Fractional or decimal values in translation vectors are handled exactly the same way as integer values:

For translation T⟨a,b⟩ where a and b are fractions or decimals:

(x, y) → (x + a, y + b)

For example:

  • Translation by ⟨0.5, -1.5⟩: (x, y) → (x + 0.5, y - 1.5)
  • Translation by ⟨1/2, 3/4⟩: (x, y) → (x + 1/2, y + 3/4)

The only difference is that the resulting coordinates may be fractional or decimal. The fundamental properties of translation (rigidity, distance preservation, etc.) remain unchanged regardless of the type of numbers in the vector.