Solved Exercises on Symmetry and Transformations in Grade 7

Master symmetry and transformations: reflection, rotation, translation, and symmetry axes through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Reflection Transformation
Exercise 1
Triangle ABC has vertices A(2, 3), B(4, 1), C(1, -1). Reflect triangle ABC across the y-axis. Find the coordinates of the reflected triangle A'B'C'.
Definition:

Reflection: A transformation that flips a figure across a line called the line of reflection. For reflection across the y-axis: (x, y) → (-x, y)

Reflection method:
  1. Identify the line of reflection
  2. Apply the reflection rule to each coordinate
  3. Plot the new points to form the reflected figure
Original Points
A(2,3), B(4,1), C(1,-1)
Reflection Rule
(x,y) → (-x,y)
Reflected Points
A'(-2,3), B'(-4,1), C'(-1,-1)
Step 1: Apply reflection rule to point A

A(2, 3) → A'(-2, 3) because (x, y) → (-x, y)

Step 2: Apply reflection rule to point B

B(4, 1) → B'(-4, 1) because (x, y) → (-x, y)

Step 3: Apply reflection rule to point C

C(1, -1) → C'(-1, -1) because (x, y) → (-x, y)

Step 4: Verify distances from y-axis

Point A is 2 units from y-axis, A' is also 2 units from y-axis but on opposite side

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

The reflected triangle A'B'C' has vertices A'(−2, 3), B'(−4, 1), C'(−1, −1)

Applied rules:

• Reflection across y-axis: (x, y) → (-x, y)

• Distance preservation: Original and reflected points are equidistant from the line of reflection

• Orientation change: Reflection reverses the orientation of the figure

2 Rotation Transformation
Exercise 2
Point P(3, 4) is rotated 90° counterclockwise around the origin. Find the coordinates of the rotated point P'.
Definition:

Rotation: A transformation that turns a figure around a fixed point called the center of rotation. For 90° CCW rotation around origin: (x, y) → (-y, x)

Original Point
P(3, 4)
Rotation Rule
(x,y) → (-y,x)
Rotated Point
P'(-4, 3)
Step 1: Identify rotation parameters

Center: Origin (0, 0), Angle: 90° counterclockwise

Step 2: Apply rotation rule

For 90° CCW rotation around origin: (x, y) → (-y, x)

P(3, 4) → P'(-4, 3)

Step 3: Verify the rotation

Distance from origin: √(3² + 4²) = √25 = 5

Distance from origin: √((-4)² + 3²) = √25 = 5 ✓

Step 4: Confirm angle of rotation

Point moved from first quadrant to second quadrant, confirming 90° counterclockwise rotation

P'(−4, 3)
Final answer:

After rotating 90° counterclockwise around the origin, P'(−4, 3)

Applied rules:

• 90° CCW rotation: (x, y) → (-y, x)

• Distance preservation: Distance from center of rotation remains constant

• Angle preservation: Shape and size of the figure remain unchanged

3 Translation Transformation
Exercise 3
Rectangle PQRS has vertices P(1, 2), Q(4, 2), R(4, 5), S(1, 5). Translate the rectangle 3 units right and 2 units down. Find the coordinates of the translated rectangle P'Q'R'S'.
Definition:

Translation: A transformation that moves every point of a figure the same distance in the same direction. Rule: (x, y) → (x+a, y+b) where (a,b) is the translation vector

Original Points
P(1,2), Q(4,2), R(4,5), S(1,5)
Translation Vector
(3, -2)
Translation Rule
(x,y) → (x+3, y-2)
Step 1: Identify translation vector

3 units right means +3 to x-coordinate, 2 units down means -2 to y-coordinate

Translation vector: (3, -2)

Step 2: Apply translation rule to each vertex

P(1, 2) → P'(1+3, 2-2) = P'(4, 0)

Q(4, 2) → Q'(4+3, 2-2) = Q'(7, 0)

R(4, 5) → R'(4+3, 5-2) = R'(7, 3)

S(1, 5) → S'(1+3, 5-2) = S'(4, 3)

Step 3: Verify shape preservation

Rectangle dimensions remain the same: width = 3 units, height = 3 units

Step 4: Check parallelism

All sides remain parallel to their original positions, confirming it's a rigid transformation

P'(4, 0), Q'(7, 0), R'(7, 3), S'(4, 3)
Final answer:

The translated rectangle P'Q'R'S' has vertices P'(4, 0), Q'(7, 0), R'(7, 3), S'(4, 3)

Applied rules:

• Translation rule: (x, y) → (x+a, y+b) where (a,b) is the translation vector

• Shape preservation: All angles and side lengths remain unchanged

• Parallelism preservation: Lines remain parallel after translation

Rules and methods, laws,...
Reflection: (x,y) → (-x,y) or (x,y) → (x,-y)
Reflection Transformation
Rotation Rules
90° CCW: (x,y) → (-y,x)
Counterclockwise rotation
Rotation Rules
180°: (x,y) → (-x,-y)
Half turn rotation
Translation Rule
(x,y) → (x+a, y+b)
Vector translation
Key definitions:

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

Rigid transformation: A transformation that preserves distances and angles (reflection, rotation, translation)

Line of symmetry: A line that divides a figure into two congruent halves that are mirror images

Complete methodology:
  1. Identify the type of transformation: Determine if it's reflection, rotation, or translation
  2. Find transformation parameters: Center of rotation, line of reflection, or translation vector
  3. Apply the transformation rule: Use the appropriate mathematical rule for the transformation
  4. Verify the result: Check that the transformed figure maintains the required properties
Tip 1: For reflections, remember that the line of reflection acts as the perpendicular bisector of the segment connecting original and reflected points.
Tip 2: For rotations, always specify the center, angle, and direction (clockwise or counterclockwise).
Tip 3: For translations, the same movement vector applies to all points in the figure.
Common errors: Confusing clockwise vs counterclockwise rotations, misapplying reflection rules, forgetting to apply transformations to all points.
Exam preparation: Memorize transformation rules, practice identifying lines of symmetry, understand the properties preserved under each transformation.
Solution: Exercises 4 to 5
4 Symmetry Axes
Exercise 4
Identify all lines of symmetry for a regular hexagon and explain why these are the only lines of symmetry.
Definition:

Line of symmetry: A line that divides a figure into two congruent parts that are mirror images of each other. A regular hexagon has 6 lines of symmetry.

Hexagon Type
Regular Hexagon
Number of Sides
6
Lines of Symmetry
6
Step 1: Understand regular hexagon properties

A regular hexagon has 6 equal sides and 6 equal angles. It's highly symmetrical.

Step 2: Identify vertex-to-vertex symmetry lines

Draw lines connecting opposite vertices: 3 lines passing through pairs of opposite vertices

Step 3: Identify side-to-side symmetry lines

Draw lines connecting midpoints of opposite sides: 3 lines passing through midpoints of opposite sides

Step 4: Verify symmetry

Each line divides the hexagon into two congruent parts that are mirror images of each other

Step 5: Explain why no more lines exist

Any other line would not divide the hexagon symmetrically due to its regular structure

A regular hexagon has 6 lines of symmetry: 3 through opposite vertices and 3 through midpoints of opposite sides
Final answer:

A regular hexagon has exactly 6 lines of symmetry: 3 lines connecting opposite vertices and 3 lines connecting midpoints of opposite sides.

Applied rules:

• Regular polygon symmetry: A regular n-gon has n lines of symmetry

• Vertex-to-vertex lines: Connect opposite vertices when n is even

• Side-to-side lines: Connect midpoints of opposite sides when n is even

5 Composite Transformations
Exercise 5
Point A(2, 3) undergoes two transformations in sequence: first a reflection across the x-axis, then a translation of 4 units right and 1 unit up. Find the final coordinates of point A''.
Definition:

Composite transformation: The combination of two or more transformations applied sequentially. Order matters in composite transformations.

Original Point
A(2, 3)
First Transformation
Reflection across x-axis
Second Transformation
Translation (4, 1)
Step 1: Apply first transformation (reflection)

Reflection across x-axis: (x, y) → (x, -y)

A(2, 3) → A'(2, -3)

Step 2: Apply second transformation (translation)

Translation of 4 units right and 1 unit up: (x, y) → (x+4, y+1)

A'(2, -3) → A''(2+4, -3+1) = A''(6, -2)

Step 3: Verify the order of operations

Transformations are applied in sequence: first reflection, then translation

Step 4: Confirm final position

Starting at (2, 3), reflected to (2, -3), then translated to (6, -2)

A''(6, -2)
Final answer:

After the composite transformation, the final coordinates are A''(6, -2)

Applied rules:

• Order of operations: Apply transformations in the given sequence

• Reflection across x-axis: (x, y) → (x, -y)

• Translation: (x, y) → (x+a, y+b)

Key Concepts, Laws, Methods, and Definitions
(x,y) → (-x,y) for reflection across y-axis
Reflection Rule
Key definitions:

Transformation: A mapping of a geometric figure that changes its position, size, or shape

Rigid transformation: A transformation that preserves distances and angles (isometry)

Line of symmetry: A line that divides a figure into two mirror-image halves

Center of rotation: The fixed point around which a figure rotates

Translation vector: The directed distance and direction of a translation

Complete methodology:
  1. Identify the transformation type: Determine if it's reflection, rotation, translation, or composite
  2. Find transformation parameters: Locate center, line, or vector as needed
  3. Apply the mathematical rule: Use the appropriate coordinate transformation
  4. Verify properties: Check that distances and angles are preserved
Tip 1: Always sketch the original and transformed figures to visualize the transformation.
Tip 2: Remember that rigid transformations preserve area, perimeter, and angle measures.
Tip 3: For composite transformations, work from left to right (apply first transformation, then second).
Tip 4: Count the number of symmetry lines systematically to avoid missing any.
Common errors: Confusing transformation rules, applying transformations in wrong order, miscounting symmetry lines, forgetting to apply transformations to all points.
Exam preparation: Memorize transformation rules, practice identifying symmetry in various shapes, understand the properties preserved under each transformation, master composite transformations.
Transformation Rules:

• Reflection across y-axis: (x, y) → (-x, y)

• Reflection across x-axis: (x, y) → (x, -y)

• 90° CCW rotation around origin: (x, y) → (-y, x)

• 180° rotation around origin: (x, y) → (-x, -y)

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

• Regular n-gon symmetry lines: n lines of symmetry

Questions & Answers

Question: I'm confused about the difference between reflection and rotation. How can I tell which one to use when looking at a figure?

Answer: Great question! Here's how to distinguish between them:

  • Reflection: Creates a mirror image across a line. The figure appears flipped over a line of reflection.
  • Rotation: Turns the figure around a fixed point. The figure maintains its orientation but changes position.

Key differences:

  • In reflection, the figure and its image are on opposite sides of the line of reflection.
  • In rotation, the figure and its image are around the center of rotation.
  • Reflection changes the orientation (clockwise becomes counterclockwise), while rotation preserves orientation.

Example: If you see a triangle and its "mirror" on the other side of a line, it's reflection. If you see a triangle turned at an angle around a point, it's rotation.

Question: How do I find all the lines of symmetry for irregular shapes? I often miss some.

Answer: Here's a systematic approach to find all lines of symmetry:

  1. Visual inspection: Look for obvious dividing lines where one half mirrors the other.
  2. Fold test: Mentally fold the shape along potential lines to see if the halves match perfectly.
  3. Vertex connections: For polygons, try connecting vertices that might form symmetry lines.
  4. Midpoint connections: Connect midpoints of potentially symmetric sides.
  5. Systematic check: Go around the shape at regular intervals to avoid missing any possibilities.

For irregular shapes, there may be no lines of symmetry, one, or rarely more. Always verify by checking if both sides of the potential line are exact mirror images.

Tip: Start with horizontal and vertical lines, then try diagonal directions.

Question: When doing composite transformations, does the order matter? Could I get different results if I changed the order?

Answer: Yes, the order of transformations absolutely matters! In most cases, changing the order will give you a different result.

Example: Let's take point (1, 2) and apply two transformations:

  • Order 1: Reflect across x-axis, then translate 3 units right
  • Step 1: (1, 2) → (1, -2) [reflection]
  • Step 2: (1, -2) → (4, -2) [translation]
  • Order 2: Translate 3 units right, then reflect across x-axis
  • Step 1: (1, 2) → (4, 2) [translation]
  • Step 2: (4, 2) → (4, -2) [reflection]

In this case, both orders gave the same result, but this is not always true. For example, rotating then translating usually gives a different result than translating then rotating.

Always follow the specified order in problems!