Solved Exercises on Rotations in High School Geometry

Master rotations: center of rotation, angle of rotation, coordinate rules, and geometric properties through these 5 detailed exercises.

Rules and methods, laws,...
\(R_{O,θ}(x,y)\)
Rotation Notation
\((x,y) → (-y,x)\) (90° CCW)
Common Rotation Rule
Rotation
Rigid Transformation
Preserves distance, angles
Center of Rotation
Fixed point
Does not move
Angle of Rotation
Measured counterclockwise
Positive = CCW, Negative = CW
Key definitions:

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

Center of Rotation: The fixed point around which the figure rotates. This point does not move during the rotation.

Angle of Rotation: The measure of rotation, typically measured in degrees counterclockwise (CCW) as positive, clockwise (CW) as negative.

Counterclockwise (CCW): The direction opposite to the rotation of clock hands.

Rotation Methods:
  1. Identify center and angle: Determine center of rotation and angle of rotation
  2. Apply coordinate rules: Use specific rules for common rotations about origin
  3. Calculate distances: Verify that distance from center remains constant
  4. Graph the result: Plot the rotated figure
  5. Confirm properties: Verify the transformation is rigid
Tip 1: Counterclockwise is positive, clockwise is negative!
Tip 2: Center of rotation stays fixed during transformation.
Tip 3: Remember common rotation rules for origin.
Tip 4: Distance from center to any point remains constant.
Solution: Exercises 1 to 3
1 90° Counterclockwise Rotation
Exercise 1
Point P(3, 2) is rotated 90° counterclockwise about the origin. Find the coordinates of P'.
Definition:

90° Counterclockwise Rotation about Origin: The transformation rule is (x, y) → (-y, x). This rotates the point one-quarter turn counterclockwise around the origin.

Rotation Method:
  1. Identify the rotation: 90° CCW about origin
  2. Apply the rule: (x, y) → (-y, x)
  3. Substitute original coordinates
  4. Calculate new coordinates
Original Point
P(3, 2)
Rotation
90° CCW about origin
Rotated Point
P'(-2, 3)
Step 1: Identify the rotation rule

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

Step 2: Apply the rule to point P(3, 2)

Substitute x = 3 and y = 2 into the rule (x, y) → (-y, x)

Step 3: Calculate the new coordinates

P(3, 2) → P'(-2, 3)

New x-coordinate: -y = -2

New y-coordinate: x = 3

Step 4: Verify the rotation

Distance from origin to P: √[3² + 2²] = √[9 + 4] = √13

Distance from origin to P': √[(-2)² + 3²] = √[4 + 9] = √13

Distances are equal ✓

P'(-2, 3)
Distance from origin preserved: √13
Final answer:

The coordinates of the rotated point are P'(-2, 3). The distance from the origin remains √13 units, confirming this is a rigid transformation.

Applied rules:

• 90° CCW Rule: (x, y) → (-y, x)

• Distance Preservation: √[x² + y²] remains constant

• Rigidity: Rotations preserve all distances and angles

2 180° Rotation
Exercise 2
Triangle ABC has vertices A(1, 2), B(4, 0), C(2, -3). Rotate 180° about the origin. Find coordinates of A'B'C'.
Definition:

180° Rotation about Origin: The transformation rule is (x, y) → (-x, -y). This rotates the point halfway around the origin, effectively reflecting across both axes.

Pre-image
A(1,2), B(4,0), C(2,-3)
Rotation
180° about origin
Image
A'(-1,-2), B'(-4,0), C'(-2,3)
Step 1: Apply 180° rotation rule to vertex A

Rule: (x, y) → (-x, -y)

A(1, 2) → A'(-1, -2)

Step 2: Apply 180° rotation rule to vertex B

B(4, 0) → B'(-4, 0)

Step 3: Apply 180° rotation rule to vertex C

C(2, -3) → C'(-2, 3)

Step 4: Verify distance preservation for one vertex

Distance OA: √[1² + 2²] = √5

Distance OA': √[(-1)² + (-2)²] = √5

Distance preserved ✓

Step 5: Confirm the rotation is rigid

Since all vertices maintain the same distance from origin, the transformation is rigid

Triangle A'B'C' is congruent to triangle ABC

A'(-1, -2), B'(-4, 0), C'(-2, 3)
Triangle A'B'C' ≅ Triangle ABC
Final answer:

The coordinates of the rotated triangle are A'(-1, -2), B'(-4, 0), and C'(-2, 3). The rotation preserved the shape and size of the triangle.

Applied rules:

• 180° Rule: (x, y) → (-x, -y)

• Uniform Rotation: Apply same rule to all points

• Congruence: Original and rotated figures are congruent

3 270° Counterclockwise Rotation
Exercise 3
Point Q(-2, 5) is rotated 270° counterclockwise about the origin. Find the coordinates of Q'.
Definition:

270° Counterclockwise Rotation about Origin: The transformation rule is (x, y) → (y, -x). This is equivalent to a 90° clockwise rotation.

Original Point
Q(-2, 5)
Rotation
270° CCW about origin
Rotated Point
Q'(5, 2)
Step 1: Identify the rotation rule

For 270° counterclockwise rotation about origin: (x, y) → (y, -x)

Step 2: Apply the rule to point Q(-2, 5)

Substitute x = -2 and y = 5 into the rule (x, y) → (y, -x)

Step 3: Calculate the new coordinates

Q(-2, 5) → Q'(5, -(-2)) = Q'(5, 2)

New x-coordinate: y = 5

New y-coordinate: -x = -(-2) = 2

Step 4: Verify with alternative approach

270° CCW = 360° - 90° = -90° (90° clockwise)

For 90° CW: (x, y) → (y, -x)

This gives the same result: (-2, 5) → (5, 2) ✓

Step 5: Confirm distance preservation

Distance from origin to Q: √[(-2)² + 5²] = √[4 + 25] = √29

Distance from origin to Q': √[5² + 2²] = √[25 + 4] = √29

Distance preserved ✓

Q'(5, 2)
Distance from origin preserved: √29
Final answer:

The coordinates of the rotated point are Q'(5, 2). The rotation preserved the distance from the origin, confirming this is a rigid transformation.

Applied rules:

• 270° CCW Rule: (x, y) → (y, -x)

• Alternative Interpretation: 270° CCW = 90° CW

• Distance Preservation: Rotation maintains distance from center

Solution: Exercises 4 to 5
4 Rotation About a Point Other Than Origin
Exercise 4
Point R(4, 1) is rotated 90° counterclockwise about point C(2, 3). Find the coordinates of R'.
Definition:

Rotation About Arbitrary Point: To rotate about a point other than the origin, translate the system so the center is at origin, apply the rotation rule, then translate back.

Original Point
R(4, 1)
Center of Rotation
C(2, 3)
Rotation
90° CCW
Result
R'(0, 5)
Step 1: Translate system so center is at origin

Translate by (-2, -3) to move C(2, 3) to origin

R(4, 1) → R₁(4-2, 1-3) = R₁(2, -2)

Step 2: Apply 90° CCW rotation about origin

Rule: (x, y) → (-y, x)

R₁(2, -2) → R₂(-(-2), 2) = R₂(2, 2)

Step 3: Translate system back

Translate by (2, 3) to move center back to C(2, 3)

R₂(2, 2) → R'(2+2, 2+3) = R'(4, 5)

Step 4: Verify the rotation

Distance from C to R: √[(4-2)² + (1-3)²] = √[4 + 4] = √8

Distance from C to R': √[(4-2)² + (5-3)²] = √[4 + 4] = √8

Distance preserved ✓

Step 5: Verify center remains fixed

Point C(2, 3) should remain unchanged after rotation about itself

Following the same steps: C(2,3) → C₁(0,0) → C₂(0,0) → C'(2,3)

Center remains fixed ✓

R'(4, 5)
Distance from C preserved: √8 units
Final answer:

The coordinates of the rotated point are R'(4, 5). The rotation preserved the distance from the center of rotation, confirming this is a rigid transformation.

Applied rules:

• Three-Step Process: Translate → Rotate → Translate Back

• Distance Preservation: Distance from center to any point remains constant

• Center Fixed: Center of rotation does not move

5 Rotation and Distance Preservation
Exercise 5
Points A(1, 0) and B(0, 1) are rotated 90° counterclockwise about the origin to A' and B'. Verify that the distance between A and B equals the distance between A' and B'.
Definition:

Rigid Transformation Property: Rotations 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(1,0), B(0,1)
Rotation
90° CCW about origin
Rotated Points
A'(0,1), B'(-1,0)
Step 1: Find coordinates of A' and B'

For 90° CCW: (x, y) → (-y, x)

A(1, 0) → A'(-0, 1) = A'(0, 1)

B(0, 1) → B'(-1, 0) = B'(-1, 0)

Step 2: Calculate distance AB using distance formula

AB = √[(0-1)² + (1-0)²] = √[(-1)² + 1²] = √[1 + 1] = √2

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

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

Step 4: Compare distances

Distance AB = √2 units

Distance A'B' = √2 units

AB = A'B' ✓

Step 5: Verify individual distance preservation

Distance from origin to A: √[1² + 0²] = 1

Distance from origin to A': √[0² + 1²] = 1

Distance preserved ✓

Step 6: Confirm rigid transformation property

Since all distances are preserved, the rotation is a rigid transformation

This confirms that rotations preserve all geometric properties

AB = A'B' = √2 units
Rotation 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 rotation preserves distances, confirming it is a rigid transformation.

Applied rules:

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

• Rigidity Property: Rotations preserve all distances

• Verification: Compare distances before and after transformation

Key Concepts: Rotations

Rotation Fundamentals

📊
Common Rotation Rules

For rotations about origin:

90° CCW: (x,y) → (-y, x)
180°: (x,y) → (-x, -y)
270° CCW: (x,y) → (y, -x)
360°: (x,y) → (x, y)
Center of Rotation

Fixed point during rotation:

R_{C,θ} where C is center

Center does not move during rotation

Properties Preserved

Rotations preserve all geometric properties:

• Distances from center

• Distance between points

• Angle measures

• Parallelism

• Congruence

Rotation Process
1
Identify Center & Angle
2
Apply Rule
3
Calculate New Coordinates
4
Verify Distance
5
Graph Result
Rotation Directions
+90° = CCW
-90° = CW
+180° = ½ turn
+270° = ¾ turn CCW
-270° = ¼ turn CW
+360° = full turn
Rigid Transformation
Preserves Distance
Preserves Angles
Preserves Shape
Preserves Size
Creates Congruence
Rotation = Turn!

Questions & Answers

Question: How do I remember the different rotation rules? They seem so random!

Answer: There's actually a pattern to rotation rules! Think about what happens to the point (1,0) under each rotation:

  • 90° CCW: (1,0) → (0,1) - x and y swap places, y gets negative sign
  • 180°: (1,0) → (-1,0) - both coordinates get negative signs
  • 270° CCW: (1,0) → (0,-1) - x and y swap places, x gets negative sign

For the general rule (x,y) → (?, ?):

  • 90° CCW: (x,y) → (-y, x)
  • 180°: (x,y) → (-x, -y)
  • 270° CCW: (x,y) → (y, -x)

Remember: 270° CCW is the same as 90° CW!

Question: What happens when I rotate a point about itself? Does it move?

Answer: When you rotate a point about itself, the point does not move! This is because the center of rotation remains fixed during any rotation.

For example, if you rotate point P(3, 4) about P(3, 4), the result is still P(3, 4).

This is a fundamental property of rotations: the center of rotation is always invariant (unchanged) under the transformation. This is why when rotating about a point other than the origin, we need to translate the system so that the center of rotation moves to the origin, perform the rotation, and then translate back.

Question: How do I handle clockwise rotations? Are there special rules?

Answer: Clockwise rotations can be converted to equivalent counterclockwise rotations:

  • 90° CW = 270° CCW → (x,y) → (y, -x)
  • 180° CW = 180° CCW → (x,y) → (-x, -y)
  • 270° CW = 90° CCW → (x,y) → (-y, x)

Alternatively, you can use the direct clockwise rules:

  • 90° CW: (x,y) → (y, -x)
  • 180° CW: (x,y) → (-x, -y)
  • 270° CW: (x,y) → (-y, x)

It's often easier to convert clockwise to counterclockwise using the relationship: θ° CW = (360° - θ)° CCW.

Question: Why are rotations considered rigid transformations? What exactly is preserved?

Answer: Rotations are rigid transformations because they preserve all metric properties of geometric figures:

  • Distances: The distance between any two points remains unchanged
  • Angles: All angle measures are preserved
  • Areas: The area of any figure remains the same
  • Shapes: The overall shape and proportions are maintained
  • Parallelism: Parallel lines remain parallel

Additionally, rotations preserve the distance from the center of rotation to any point, and they maintain the orientation of figures (unlike reflections). This is why rotations produce congruent figures - the transformed figure is identical to the original in all measurable ways except position and orientation.