Solved Exercises on Translations in Grade 8

Master translations: geometric transformations, coordinate rules, and problem-solving through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Translation Rule
Exercise 1
Translate point A(3, -2) by the rule (x, y) → (x+4, y-3). Find the coordinates of the image point A'.
Definition:

Translation: A geometric transformation that moves every point of a figure the same distance in the same direction. The rule (x, y) → (x+a, y+b) translates a point a units horizontally and b units vertically.

Solution method:
  1. Identify the translation rule: (x, y) → (x+a, y+b)
  2. Identify the original coordinates: (x, y)
  3. Apply the rule by adding the translation values to the original coordinates
  4. Write the new coordinates as the image point
Given
A(3,-2), T:(x,y)→(x+4,y-3)
Apply rule
A'(3+4, -2-3)
Solution
A'(7, -5)
Step 1: Identify the translation rule

The rule is (x, y) → (x+4, y-3)

This means move 4 units right and 3 units down

Step 2: Identify original coordinates

Original point A has coordinates (3, -2)

Step 3: Apply the translation rule

For x-coordinate: 3 + 4 = 7

For y-coordinate: -2 + (-3) = -5

Step 4: Write the image coordinates

The translated point A' has coordinates (7, -5)

A' = (7, -5)
Final answer:

The coordinates of A' after translation are (7, -5)

Applied rules:

• Translation rule: (x, y) → (x+a, y+b) moves a units horizontally, b units vertically

• Positive x: Move right

• Negative y: Move down

• Verification: Check that A moved 4 units right and 3 units down to reach A'

2 Translating a Triangle
Exercise 2
Triangle ABC has vertices A(1, 2), B(4, 1), and C(2, 5). Translate the triangle by the rule (x, y) → (x-3, y+2). Find the coordinates of the image triangle A'B'C'.
Definition:

Translation of a figure: Apply the same translation rule to each vertex of the figure to obtain the image figure.

Rule
(x,y)→(x-3,y+2)
Apply to each vertex
A', B', C' calculated
Solution
A'(-2,4), B'(1,3), C'(-1,7)
Step 1: Identify the translation rule

The rule is (x, y) → (x-3, y+2)

This means move 3 units left and 2 units up

Step 2: Apply rule to vertex A(1, 2)

A': (1-3, 2+2) = (-2, 4)

Step 3: Apply rule to vertex B(4, 1)

B': (4-3, 1+2) = (1, 3)

Step 4: Apply rule to vertex C(2, 5)

C': (2-3, 5+2) = (-1, 7)

A'(-2, 4), B'(1, 3), C'(-1, 7)
Final answer:

The coordinates of the image triangle A'B'C' are A'(-2, 4), B'(1, 3), and C'(-1, 7)

Applied rules:

• Translation rule: Apply the same rule to each vertex

• Figure preservation: Shape, size, and orientation remain unchanged

• Parallel movement: All points move in the same direction and distance

• Verification: Check that each vertex moved according to the rule

3 Real-World Application
Exercise 3
A robot starts at position R(5, 3) on a coordinate grid. It moves 6 units east and 4 units north to reach position R'. What are the coordinates of R'?
Definition:

Directional translation: East corresponds to positive x-direction, North corresponds to positive y-direction.

Start
R(5,3)
Movement
6 units E, 4 units N
Result
R'(11, 7)
Step 1: Identify starting position

Robot starts at R(5, 3)

Step 2: Interpret directional movements

Move 6 units east = move 6 units in positive x direction

Move 4 units north = move 4 units in positive y direction

Step 3: Apply translation rule

The movement is equivalent to the rule (x, y) → (x+6, y+4)

Step 4: Calculate new coordinates

R': (5+6, 3+4) = (11, 7)

R' = (11, 7)
Final answer:

The coordinates of R' are (11, 7)

Applied rules:

• Direction mapping: East = +x, West = -x, North = +y, South = -y

• Translation rule: (x, y) → (x+a, y+b)

• Verification: R moved 6 units right and 4 units up to reach R'

Rules and methods, laws,...
(x, y) → (x + a, y + b)
Translation Rule
T_{(a,b)}: (x,y) → (x+a, y+b)
Notation
(x', y') = (x+a, y+b)
Image Coordinates
Right Translation
a > 0
Add to x-coordinate
Left Translation
a < 0
Subtract from x-coordinate
Up Translation
b > 0
Add to y-coordinate
Down Translation
b < 0
Subtract from y-coordinate
Key definitions:

Translation: A rigid transformation that slides a figure along a straight line without changing its size, shape, or orientation

Transformation: A change in the position, size, or shape of a geometric figure

Pre-image: The original figure before the transformation

Image: The figure after the transformation

Translation vector: The directed line segment that describes the translation (a, b)

Rigid motion: A transformation that preserves distance and angle measures

Isometry: Another term for rigid motion, preserving all geometric properties

Coordinate notation: (x, y) → (x+a, y+b) describes the transformation rule

Complete methodology:
  1. Identify the translation rule: Determine (x, y) → (x+a, y+b) from the given information
  2. Understand the movement: Positive a = right, negative a = left; positive b = up, negative b = down
  3. Identify pre-image points: List the coordinates of all points in the original figure
  4. Apply the rule: Add a to each x-coordinate and b to each y-coordinate
  5. Record image points: Write the new coordinates as the transformed figure
  6. Verify the transformation: Check that the figure has maintained its shape and size
Tip 1: Remember: positive x is right, negative x is left; positive y is up, negative y is down.
Tip 2: Translation preserves all geometric properties: side lengths, angles, area, and shape.
Tip 3: Each point moves the same distance in the same direction.
Tip 4: Apply the same rule to every vertex of a geometric figure.
Common errors: Mixing up x and y directions, applying different rules to different points, forgetting to apply the rule to all points of a figure, confusing translation with other transformations.
Exam preparation: Practice with various translation rules, work on translating complex figures, understand directional language, master coordinate notation.
Formulas to know by heart:

• Translation rule: (x, y) → (x+a, y+b)

• Vector notation: T_{(a,b)}(x, y) = (x+a, y+b)

• Directional mapping: East = +x, West = -x, North = +y, South = -y

• Translation preserves: distance, angle measures, parallelism, perpendicularity

• Image coordinates: (x', y') = (x+a, y+b)

Solution: Exercises 4 to 5
4 Finding Translation Rule
Exercise 4
Point P(2, -1) is translated to P'(6, 3). Find the translation rule that maps P to P'.
Definition:

Reverse translation: Given a pre-image and its image, find the translation rule by determining the change in coordinates.

Given
P(2,-1) → P'(6,3)
Find changes
Δx = 4, Δy = 4
Rule
(x,y) → (x+4,y+4)
Step 1: Identify pre-image and image

Pre-image: P(2, -1)

Image: P'(6, 3)

Step 2: Calculate change in x-coordinate

Change in x: 6 - 2 = 4

This means move 4 units right

Step 3: Calculate change in y-coordinate

Change in y: 3 - (-1) = 3 + 1 = 4

This means move 4 units up

Step 4: Write the translation rule

Since x increases by 4 and y increases by 4, the rule is:

(x, y) → (x+4, y+4)

Step 5: Verify the rule

Apply to P(2, -1): (2+4, -1+4) = (6, 3) = P' ✓

Rule: (x, y) → (x+4, y+4)
Final answer:

The translation rule is (x, y) → (x+4, y+4)

Applied rules:

• Rule determination: Find differences between corresponding coordinates

• Verification: Apply rule to original point to ensure it produces the image

• Consistency: Each coordinate changes by the same amount for all points

5 Composition of Translations
Exercise 5
Point Q(1, 4) undergoes two translations: first (x, y) → (x-2, y+3), then (x, y) → (x+5, y-1). Find the final coordinates of Q''.
Definition:

Composition of translations: Applying multiple translations sequentially. The overall effect is equivalent to a single translation.

First translation
Q(1,4) → Q'(−1,7)
Second translation
Q'(−1,7) → Q''(4,6)
Net result
Q(1,4) → Q''(4,6)
Step 1: Apply first translation to Q(1, 4)

Rule: (x, y) → (x-2, y+3)

Q': (1-2, 4+3) = (-1, 7)

Step 2: Apply second translation to Q'(-1, 7)

Rule: (x, y) → (x+5, y-1)

Q'': (-1+5, 7-1) = (4, 6)

Step 3: Verify using combined rule

Total translation: (x, y) → (x-2+5, y+3-1) = (x+3, y+2)

Q to Q'': (1+3, 4+2) = (4, 6) ✓

Q'' = (4, 6)
Final answer:

The final coordinates of Q'' are (4, 6)

Applied rules:

• Sequential application: Apply each translation one after another

• Translation combination: Net effect is sum of individual translations

• Verification: Combined rule should produce the same result

Key Concepts, Laws, Methods, and Formulas for Translations
(x, y) → (x + a, y + b)
Translation Rule
Key definitions:

Translation: A rigid transformation that moves every point of a figure the same distance in the same direction without changing its size, shape, or orientation

Rigid transformation: A transformation that preserves distances and angle measures between points

Isometry: Another term for rigid transformation; maintains all geometric properties

Pre-image: The original figure before transformation, denoted with original letters (A, B, C, etc.)

Image: The figure after transformation, denoted with primes (A', B', C', etc.)

Translation vector: The directed line segment that describes the direction and magnitude of the translation, represented as (a, b)

Coordinate notation: The mathematical representation of a translation as (x, y) → (x+a, y+b)

Composition: The process of applying multiple transformations in sequence

Parallel lines: Lines that maintain the same distance apart and never intersect

Complete methodology:
  1. Identify the transformation type: Confirm that the problem involves translation (sliding motion)
  2. Determine the translation rule: Extract the rule from given information, whether explicitly stated or implied by movement directions
  3. Understand directional components: Recognize that positive x = right, negative x = left, positive y = up, negative y = down
  4. Identify all points to transform: List all coordinates of the pre-image figure that need to be translated
  5. Apply the translation rule systematically: Add the x-component to each x-coordinate and the y-component to each y-coordinate
  6. Record the image coordinates: Write the new coordinates using prime notation (A' for the image of A)
  7. Verify the transformation: Check that the figure maintains its shape, size, and orientation
  8. Interpret results in context: Apply findings to real-world scenarios when applicable
Tip 1: Remember the coordinate direction rules: (+x) moves right, (-x) moves left, (+y) moves up, (-y) moves down.
Tip 2: Translation preserves all geometric properties: side lengths, angles, area, parallelism, and perpendicularity.
Tip 3: Every point of the figure moves the same distance in the same direction during translation.
Tip 4: When finding a translation rule, subtract the pre-image coordinates from the image coordinates.
Tip 5: For compositions of translations, add the vectors to find the net translation.
Tip 6: Always verify your results by checking that the transformation preserves geometric properties.
Common errors: Mixing up x and y directions, applying different rules to different points of the same figure, forgetting to apply the rule to all points, confusing translation with rotation or reflection, misinterpreting directional language.
Memory aids: "Slide the same way" for all points, "Rigid motion keeps everything the same except position", "Plus moves positive direction, minus moves negative direction".
Problem-solving strategies: Draw the original and translated figures to visualize the transformation, check that distances and angles are preserved, use the translation rule consistently for all points.
Essential formulas and theorems:

• Translation rule: (x, y) → (x+a, y+b) where a is horizontal shift and b is vertical shift

• Vector notation: T_{(a,b)}(x, y) = (x+a, y+b)

• Directional mapping: East = +x, West = -x, North = +y, South = -y

• Inverse translation: If (x, y) → (x+a, y+b), then (x', y') → (x'-a, y'-b)

• Composition of translations: T_{(a,b)} followed by T_{(c,d)} equals T_{(a+c, b+d)}

• Preservation properties: Distance, angle measures, parallelism, perpendicularity, area, shape

• Coordinate change: Δx = x₂ - x₁, Δy = y₂ - y₁

Visual Representation: Translation Effects
Exercise 6: Translation Vector Analysis
Visual representation of how different translation vectors affect point positions:
- Horizontal translations
- Vertical translations
- Combined translations
- Directional analysis

Analysis: The chart illustrates how different translation vectors move points in various directions.

  • Positive x-components move points right
  • Positive y-components move points up
  • Combined vectors create diagonal movement
  • The magnitude of the vector determines the distance moved

Questions & Answers

Question: How do I know which direction to move when I see a translation rule like (x, y) → (x-3, y+2)?

Answer: Here's how to interpret translation rules:

  • For x-coordinate changes: Positive value means move right, negative value means move left
  • For y-coordinate changes: Positive value means move up, negative value means move down

In the rule (x, y) → (x-3, y+2):

  • 'x-3' means move 3 units to the left (negative x direction)
  • 'y+2' means move 2 units up (positive y direction)

Remember: x corresponds to left-right movement, y corresponds to up-down movement.

For example, point (5, 4) under this rule becomes (5-3, 4+2) = (2, 6).

Question: Does translation change the size or shape of a figure?

Answer: No, translation does NOT change the size or shape of a figure. Translation is a "rigid transformation" or "isometry," which means:

  • Side lengths remain the same
  • Angle measures remain the same
  • Area remains the same
  • Shape remains the same
  • Parallel lines remain parallel
  • Perpendicular lines remain perpendicular

Translation only changes the position of the figure. It's like sliding the entire figure without stretching, shrinking, rotating, or flipping it.

This is different from other transformations like dilation, which does change the size of a figure.

Question: How do I find the translation rule if I'm given a point and its image?

Answer: To find the translation rule when given a point and its image:

Method: Subtract the pre-image coordinates from the image coordinates

  • Change in x = x-image - x-preimage
  • Change in y = y-image - y-preimage

Example: If point A(2, 5) maps to A'(7, 1), find the rule:

  • Change in x: 7 - 2 = 5
  • Change in y: 1 - 5 = -4
  • Translation rule: (x, y) → (x+5, y-4)

Verification: Apply the rule to the original point: (2+5, 5-4) = (7, 1) ✓

This method works for any pre-image and image pair.

Question: What happens when I apply multiple translations to a point?

Answer: When applying multiple translations, the result is equivalent to a single translation. Here's how it works:

Process: Apply each translation sequentially (one after another)

Example: Point P(1, 2) undergoes T₁: (x,y) → (x+3, y-1) then T₂: (x,y) → (x-2, y+4)

  • After T₁: (1+3, 2-1) = (4, 1)
  • After T₂: (4-2, 1+4) = (2, 5)

Net result: The combined effect is equivalent to T_net: (x,y) → (x+1, y+3)

  • Check: (1+1, 2+3) = (2, 5) ✓

To find the net translation, add the individual translation vectors: (3, -1) + (-2, 4) = (1, 3).

Question: How can I check if my translation is correct?

Answer: Here are several ways to verify your translation:

  1. Rule verification: Apply the translation rule to the original point and confirm it produces your answer
  2. Distance preservation: Measure distances between points in the original figure and compare to the translated figure
  3. Visual inspection: On a graph, check that the figure moved the expected distance in the correct direction
  4. Coordinate differences: Ensure all points moved by the same x and y amounts
  5. Shape preservation: Verify that angles and side lengths remain unchanged

Example: If translating A(1,2) by (x,y)→(x+3,y-2) to get A'(4,0):

  • Check: (1+3, 2-2) = (4, 0) ✓
  • Distance moved: 3 units right, 2 units down ✓

Always double-check that the transformation maintains geometric properties.