Solved Exercises on Rotations in Grade 8

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

Solution: Exercises 1 to 3
1 90° Counterclockwise Rotation
Exercise 1
Rotate point A(3, 4) 90° counterclockwise about the origin. Find the coordinates of the image point A'.
Definition:

90° counterclockwise rotation about origin: A transformation that rotates a point 90° in the counterclockwise direction around the origin (0,0). The rule is (x, y) → (-y, x).

Solution method:
  1. Identify the rotation: 90° counterclockwise about origin
  2. Apply the rule: (x, y) → (-y, x)
  3. Replace x with -y and y with x
  4. Write the new coordinates
Given
A(3,4)
Rule
(x,y)→(-y,x)
Solution
A'(-4, 3)
Step 1: Identify the rotation rule

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

This means replace x with -y and y with x

Step 2: Identify original coordinates

Original point A has coordinates (3, 4)

Step 3: Apply the rotation rule

For x-coordinate: -y = -4

For y-coordinate: x = 3

Step 4: Write the image coordinates

The rotated point A' has coordinates (-4, 3)

A' = (-4, 3)
Final answer:

The coordinates of A' after 90° counterclockwise rotation about the origin are (-4, 3)

Applied rules:

• 90° CCW rotation: (x, y) → (-y, x)

• Distance preservation: A and A' are equidistant from the origin

• Angle preservation: The angle between original and rotated segments is 90°

• Verification: A and A' maintain the same distance from origin

2 180° Rotation
Exercise 2
Triangle ABC has vertices A(2, -1), B(4, 3), and C(-1, 5). Rotate the triangle 180° about the origin. Find the coordinates of the image triangle A'B'C'.
Definition:

180° rotation about origin: A transformation that rotates a point 180° around the origin (0,0). The rule is (x, y) → (-x, -y), which changes the signs of both coordinates.

Rule
(x,y)→(-x,-y)
Apply to each vertex
A', B', C' calculated
Solution
A'(-2,1), B'(-4,-3), C'(1,-5)
Step 1: Identify the rotation rule

For 180° rotation about origin: (x, y) → (-x, -y)

This means change the sign of both coordinates

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

A': (-(2), -(-1)) = (-2, 1)

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

B': (-(4), -(3)) = (-4, -3)

Step 4: Apply rule to vertex C(-1, 5)

C': (-(-1), -(5)) = (1, -5)

A'(-2, 1), B'(-4, -3), C'(1, -5)
Final answer:

The coordinates of the image triangle A'B'C' are A'(-2, 1), B'(-4, -3), and C'(1, -5)

Applied rules:

• 180° rotation: (x, y) → (-x, -y)

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

• Distance preservation: Each point and its image are equidistant from origin

• Verification: Each vertex moved according to the rule

3 90° Clockwise Rotation
Exercise 3
Rotate point P(5, -2) 90° clockwise about the origin. Find the coordinates of the image point P'.
Definition:

90° clockwise rotation about origin: A transformation that rotates a point 90° in the clockwise direction around the origin (0,0). The rule is (x, y) → (y, -x).

Given
P(5,-2)
Rule
(x,y)→(y,-x)
Solution
P'(-2, -5)
Step 1: Identify the rotation rule

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

This means replace x with y and y with -x

Step 2: Identify original coordinates

Original point P has coordinates (5, -2)

Step 3: Apply the rotation rule

For x-coordinate: y = -2

For y-coordinate: -x = -5

P' = (-2, -5)
Final answer:

The coordinates of P' after 90° clockwise rotation about the origin are (-2, -5)

Applied rules:

• 90° CW rotation: (x, y) → (y, -x)

• Distance preservation: P and P' are equidistant from the origin

• Angle preservation: The angle between original and rotated segments is 90°

• Verification: P and P' maintain the same distance from origin

Rules and methods, laws,...
(x, y) → (-y, x)
90° CCW rotation
(x, y) → (-x, -y)
180° rotation
(x, y) → (y, -x)
90° CW rotation
90° CCW
(x,y) → (-y,x)
Counterclockwise rotation
90° CW
(x,y) → (y,-x)
Clockwise rotation
180°
(x,y) → (-x,-y)
Half turn
270° CCW
(x,y) → (y,-x)
Equivalent to 90° CW
Key definitions:

Rotation: A rigid transformation that turns a figure around a fixed point called the center of rotation

Center of rotation: The fixed point around which the figure is rotated

Angle of rotation: The measure of the angle through which the figure is turned

Pre-image: The original figure before the transformation

Image: The figure after the transformation

Rigid transformation: A transformation that preserves distance and angle measures

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

Counterclockwise: The direction opposite to the rotation of clock hands

Clockwise: The direction of rotation of clock hands

Coordinate notation: Mathematical representation of the transformation rule

Complete methodology:
  1. Identify the center of rotation: Determine the point around which the figure rotates (often the origin)
  2. Determine the angle and direction: Identify the degree measure and direction (clockwise or counterclockwise)
  3. Select the appropriate rule: Choose the correct transformation rule based on the angle and direction
  4. Identify pre-image points: List the coordinates of all points in the original figure
  5. Apply the transformation rule: Transform each point according to the rotation rule
  6. Record image points: Write the new coordinates as the rotated figure
  7. Verify the transformation: Check that the figure has maintained its shape and size, and that the rotation is correct
Tip 1: Remember: 90° CCW is (-y, x), 90° CW is (y, -x), 180° is (-x, -y).
Tip 2: Rotation preserves all geometric properties: side lengths, angles, area, and shape.
Tip 3: 270° CCW is the same as 90° CW, and vice versa.
Tip 4: Each point and its image are equidistant from the center of rotation.
Common errors: Confusing clockwise and counterclockwise directions, applying wrong rotation rule, forgetting to change signs, confusing rotation with reflection, not applying the rule to all points of a figure.
Exam preparation: Memorize key rotation rules, practice with various angles, understand coordinate changes, work on rotating complex figures.
Formulas to know by heart:

• 90° counterclockwise: (x, y) → (-y, x)

• 90° clockwise: (x, y) → (y, -x)

• 180° rotation: (x, y) → (-x, -y)

• 270° counterclockwise: (x, y) → (y, -x)

• 270° clockwise: (x, y) → (-y, x)

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

Solution: Exercises 4 to 5
4 Finding Rotation Information
Exercise 4
Point A(2, 3) is rotated to A'(-3, 2). What is the angle and direction of rotation about the origin?
Definition:

Rotation identification: Determining the angle and direction of rotation by comparing the original point and its image.

Given
A(2,3) → A'(-3,2)
Compare coordinates
(x,y) → (-y,x)
Rotation
90° CCW
Step 1: Compare pre-image and image coordinates

A(2, 3) and A'(-3, 2)

Original x-coordinate (2) became new y-coordinate (2)

Original y-coordinate (3) became negative of new x-coordinate (-(-3) = 3)

Step 2: Identify the pattern

From (x, y) to (-y, x) is the pattern for 90° counterclockwise rotation

Step 3: Verify the rotation rule

90° CCW: (x, y) → (-y, x)

A(2, 3) → (-3, 2) = A' ✓

Step 4: Confirm the rotation

Both points are equidistant from origin: √(2² + 3²) = √13 = √((-3)² + 2²)

90° counterclockwise rotation
Final answer:

The rotation is 90° counterclockwise about the origin.

Applied rules:

• Pattern recognition: Identify which coordinates change

• Rule identification: Match pattern to known rotation rules

• Verification: Check that points are equidistant from center of rotation

5 Multiple Rotations
Exercise 5
Point P(1, 2) undergoes two rotations: first 90° CCW, then 90° CCW again. Find the final coordinates of P''.
Definition:

Composition of rotations: Applying multiple rotations sequentially. The overall effect may be equivalent to a single rotation.

First rotation
P(1,2) → P'(-2,1)
Second rotation
P'(-2,1) → P''(-1,-2)
Net result
180° rotation
Step 1: Apply first rotation (90° CCW)

Rule: (x, y) → (-y, x)

P': (1, 2) → (-2, 1)

Step 2: Apply second rotation (90° CCW)

Rule: (x, y) → (-y, x)

P'': (-2, 1) → (-1, -2)

Step 3: Verify using combined effect

Two 90° CCW rotations = one 180° rotation

180° rotation: (x, y) → (-x, -y)

P(1, 2) → (-1, -2) ✓

P'' = (-1, -2)
Final answer:

The final coordinates of P'' are (-1, -2)

Applied rules:

• Sequential application: Apply each rotation one after another

• Composition result: Two 90° rotations = one 180° rotation

• Verification: Check that the final result matches the expected transformation

Key Concepts, Laws, Methods, and Formulas for Rotations
(x, y) → (-y, x)
90° CCW rotation
Key definitions:

Rotation: A rigid transformation that turns a figure around a fixed point called the center of rotation, maintaining the same distance from the center for all points

Center of rotation: The fixed point around which the figure rotates, typically the origin (0, 0) unless specified otherwise

Angle of rotation: The measure of the angle through which the figure is turned, measured in degrees

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

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

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

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

Counterclockwise rotation: Rotation in the opposite direction to the hands of a clock (positive direction)

Clockwise rotation: Rotation in the same direction as the hands of a clock (negative direction)

Coordinate notation: The mathematical representation of a rotation as (x, y) → (new x, new y)

Complete methodology:
  1. Identify the transformation type: Confirm that the problem involves rotation (turning around a point)
  2. Determine the center of rotation: Identify the point around which the figure rotates (often the origin)
  3. Identify the angle and direction: Determine the degree measure and direction (clockwise or counterclockwise)
  4. Select the appropriate rule: Choose the correct transformation rule based on the angle and direction
  5. Identify all points to rotate: List all coordinates of the pre-image figure that need to be transformed
  6. Apply the rotation rule systematically: Transform each coordinate according to the specific rule for that rotation
  7. Record the image coordinates: Write the new coordinates using prime notation (A' for the image of A)
  8. Verify the transformation: Check that the figure maintains its shape, size, and that the rotation is correct
  9. Interpret results in context: Apply findings to real-world scenarios when applicable
Tip 1: Remember: 90° CCW is (-y, x), 90° CW is (y, -x), 180° is (-x, -y).
Tip 2: Rotation preserves all geometric properties: side lengths, angles, area, parallelism, and perpendicularity.
Tip 3: 270° CCW is equivalent to 90° CW, and 270° CW is equivalent to 90° CCW.
Tip 4: Each point and its image are equidistant from the center of rotation.
Tip 5: Two 90° rotations equal one 180° rotation.
Tip 6: Always verify your results by checking that the transformation preserves geometric properties.
Common errors: Confusing clockwise and counterclockwise directions, applying wrong rotation rule, forgetting to change signs, confusing rotation with reflection or translation, misidentifying the center of rotation, not applying the rule to all points of a figure.
Memory aids: "CCW 90°: X goes to negative Y, Y goes to X", "CW 90°: X goes to Y, Y goes to negative X", "180°: both signs change".
Problem-solving strategies: Draw the original and rotated figures to visualize the transformation, check that distances to the center of rotation are equal, verify that the transformation preserves geometric properties.
Essential formulas and theorems:

• 90° counterclockwise: (x, y) → (-y, x)

• 90° clockwise: (x, y) → (y, -x)

• 180° rotation: (x, y) → (-x, -y)

• 270° counterclockwise: (x, y) → (y, -x)

• 270° clockwise: (x, y) → (-y, x)

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

• Composition result: Two 90° rotations = one 180° rotation

Visual Representation: Rotation Effects
Exercise 6: Different Rotation Angles
Visual representation of how different rotation angles affect point positions:
- 90° CCW
- 180°
- 270° CCW
- 90° CW

Analysis: The chart illustrates how different rotation angles transform point positions.

  • 90° CCW moves point to (-y, x)
  • 180° moves point to (-x, -y)
  • 270° CCW moves point to (y, -x)
  • 90° CW moves point to (y, -x)

Questions & Answers

Question: How do I remember the coordinate changes for different rotation angles?

Answer: Here are memory aids for different rotations about the origin:

  • 90° CCW: "X goes to negative Y, Y goes to X" → (x, y) → (-y, x)
  • 90° CW: "X goes to Y, Y goes to negative X" → (x, y) → (y, -x)
  • 180°: "Both signs change" → (x, y) → (-x, -y)
  • 270° CCW: Same as 90° CW → (x, y) → (y, -x)

You can also think of it as moving the coordinate system:

  • 90° CCW: old x-axis becomes new y-axis, old y-axis becomes negative x-axis
  • 90° CW: old x-axis becomes negative y-axis, old y-axis becomes new x-axis

Practice with a few examples to internalize these patterns.

Question: Does rotation change the size or orientation of a figure?

Answer: Rotation does NOT change the size of a figure, but it DOES change the orientation:

  • Size preservation: Side lengths, angles, area, and shape remain the same
  • Position change: The figure is turned to a different position
  • Distance preservation: All distances are maintained
  • Angle preservation: All angle measures remain the same
  • Orientation change: The figure appears in a different direction

For example, if a triangle has vertices labeled clockwise, its rotation will still have vertices labeled clockwise, but the triangle will be facing a different direction.

This is different from reflection (which reverses orientation) and dilation (which changes size).

Question: How can I find the angle of rotation if I'm given a point and its image?

Answer: To find the angle of rotation:

Method 1 - Pattern Recognition:

  • Compare the coordinates of the point and its image
  • Identify which coordinates changed and how
  • Match the pattern to known rotation rules

Method 2 - Geometric Analysis:

  • Draw lines from the center of rotation to both points
  • Measure the angle between these lines
  • Determine the direction of rotation

Example: If A(2, 3) → A'(-3, 2), notice that (x, y) went to (-y, x), which is the pattern for 90° CCW rotation.

Question: What happens when I apply two rotations in a row?

Answer: The result depends on the angles and directions of the rotations:

Two rotations about the same center: The result is equivalent to a single rotation whose angle is the sum of the two rotation angles

Examples:

  • Two 90° CCW rotations = one 180° rotation
  • 90° CCW + 90° CW = 0° (identity transformation)
  • 180° + 180° = 360° (identity transformation)
  • 90° CCW + 180° = 270° CCW

Note: The order matters when rotations are about different centers, but when about the same center, the result depends only on the sum of the angles.

This is a fundamental concept in transformation geometry and group theory.

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

Answer: Here are several verification methods:

  1. Rule verification: Apply the rotation rule to the original point and confirm it produces your answer
  2. Distance check: Ensure the original point and its image are equidistant from the center of rotation
  3. Angle check: Verify that the angle between the original and rotated segments equals the rotation angle
  4. Visual inspection: On a graph, check that the figure appears rotated correctly
  5. Coordinate verification: Confirm that the appropriate coordinates changed according to the rule

Example: If rotating A(3, 4) 90° CCW to get A'(-4, 3):

  • Rule check: (3, 4) → (-4, 3) ✓
  • Distance check: Both points are √(3²+4²) = 5 units from origin ✓
  • Coordinate check: x→-y, y→x ✓

Always double-check that the transformation maintains geometric properties.