Symmetry and Transformation

Q: What is reflectional symmetry and what is the line of symmetry?

A shape has reflectional symmetry if a straight line (the axis or line of symmetry ) divides it into two halves that are mirror images of each other.

Q: What is rotational symmetry and how is its order determined?

A shape has rotational symmetry if it looks identical to its original position after being rotated around a central point by less than a full turn (\(360^\circ\)); its order of rotational symmetry is the number of times it matches during a full \(360^\circ\) turn.

Q: What is a translation and how do coordinates change when translating a point?

A translation is a slide of every point of a shape by the same distance and direction; a point \((x, y)\) shifted \(a\) units horizontally and \(b\) units vertically becomes \((x+a, y+b)\).

Q: What are the coordinate rules for reflecting a point \((x, y)\) across the \(x\)-axis and the \(y\)-axis?

Reflecting across the \(x\)-axis produces \((x, -y)\), while reflecting across the \(y\)-axis produces \((-x, y)\).

Q: How do coordinates change when a point \((x, y)\) is rotated anti-clockwise about the origin by \(90^\circ\), \(180^\circ\), and \(270^\circ\)?

About the origin, \(90^\circ\) anti-clockwise gives \((-y, x)\), \(180^\circ\) gives \((-x, -y)\), and \(270^\circ\) anti-clockwise gives \((y, -x)\).

Q: How does an enlargement or reduction with scale factor \(k\) about the origin transform the coordinates and size of a shape?

Every coordinate \((x, y)\) becomes \((kx, ky)\), where \(k > 1\) enlarges the shape and \(0

Q: Which geometric properties remain invariant under translations, reflections, rotations, and enlargements?

Translations, reflections, and rotations preserve side lengths, angles, and area (congruence); enlargements/reductions preserve angle sizes and shape proportions (similarity) but alter side lengths and area.

Q: What geometric condition is required for shapes to form a tessellation?

The sum of the interior angles of all shapes meeting at each shared vertex must equal exactly \(360^\circ\).

Practice

Mathematics · Junior Secondary · 10 min read · Updated: 17 Aug 2026

Welcome to the World of Symmetry and Transformation!

Hey everyone! Get ready to explore one of the most visual and creative parts of mathematics: Symmetry and Transformation. You see these ideas everywhere – in art, nature, video games, and even in your own reflection!

In this chapter, you'll learn how shapes can be flipped, slid, turned, and resized. It's like being a graphic designer or an animator, using maths to control how objects move and change. Don't worry if it sounds complicated; we'll break it down into simple, easy steps. Let's get started!

Part 1: Symmetry - The Beauty of Balance

What is Symmetry?

Symmetry is when a shape or object has a sense of perfect balance. If you move it in a certain way (like flipping or turning it), it looks exactly the same as it did before. It's what makes a butterfly's wings or a starfish so pleasing to look at.

Two Main Types of Symmetry

Reflectional Symmetry (or Line Symmetry)

This is the one you're probably most familiar with. A shape has reflectional symmetry if you can draw a straight line through it and each side is a mirror image of the other.

This special line is called the axis of symmetry or line of symmetry.

- Example: A heart has one line of symmetry, right down the middle.
- Example: A square has four lines of symmetry (can you find them all?).
- Example: The letter 'A' has one vertical line of symmetry, while 'E' has one horizontal line of symmetry.

Rotational Symmetry

A shape has rotational symmetry if you can turn it around a central point and it looks the same before you've completed a full 360-degree turn.

The number of times a shape looks the same during a full turn is called its order of rotational symmetry.

- Example: A square looks the same 4 times when you rotate it around its centre. So, it has a rotational symmetry of order 4.
- Example: A pinwheel or a starfish also has rotational symmetry.

Key Takeaway for Symmetry

Symmetry is all about balance. Reflectional symmetry is a mirror image across a line. Rotational symmetry is when a shape looks the same after being turned around a point.

Part 2: Transformations - Moving Shapes Around

A transformation is a way of changing the position, orientation, or size of a shape. Think of it as giving a shape instructions on how to move or change.

Key Terms to Know:
- Object: The original shape before it's transformed.
- Image: The new shape after the transformation.

Transformations like translation, reflection, and rotation do not change the shape's size or lengths. The image is congruent to the object. Transformations like enlargement or reduction change the size, making the image similar to the object.

First, A Quick Refresher: The Coordinate Plane

Remember the coordinate plane? It's the grid with two main lines: the horizontal x-axis and the vertical y-axis. The point where they meet is called the origin, which is at coordinate \((0, 0)\).

We find any point on the grid using its coordinates, written as \((x, y)\). Just remember: "You have to walk along the hall (x-axis) before you can go up or down the stairs (y-axis)."

Transformation 1: Translation (The Slide)

A translation is just a "slide". You move every single point of a shape the exact same distance and in the exact same direction.

Real-world analogy: Imagine sliding a chess piece across the board. It doesn't flip or turn; it just slides to a new square.

How to Translate a Point

Instructions for a translation are given as movements left/right and up/down.

- Moving right adds to the x-coordinate.
- Moving left subtracts from the x-coordinate.
- Moving up adds to the y-coordinate.
- Moving down subtracts from the y-coordinate.

If a point is at \((x, y)\) and we translate it \(a\) units horizontally and \(b\) units vertically, the new point (the image) will be at \((x+a, y+b)\).

Step-by-Step Example:

Let's translate the point A(2, 1) by '4 units right and 3 units up'.

1. Start with the original x-coordinate: 2
2. Move 4 units right: \(2 + 4 = 6\). The new x-coordinate is 6.
3. Start with the original y-coordinate: 1
4. Move 3 units up: \(1 + 3 = 4\). The new y-coordinate is 4.
5. The image point A' is at (6, 4).

Common Mistake to Avoid!

Be careful with negative directions! Moving 'left' means subtracting from x, and moving 'down' means subtracting from y. Don't mix them up!

Key Takeaway for Translation

Translation is a slide. Just add or subtract from the x and y coordinates to find the new position of the image.

Transformation 2: Reflection (The Flip)

A reflection is a "flip". It's like looking in a mirror. Every point in the image is the same distance from the mirror line as the corresponding point in the object.

The mirror line is called the line of reflection.

Reflecting Across the x-axis

When you reflect a point across the horizontal x-axis, it's like the x-axis is a puddle. The point 'jumps' to the other side, but stays the same distance away from the edge of the puddle.

The Rule: To reflect a point \((x, y)\) across the x-axis, the image is \((x, -y)\).
Basically, the x-coordinate stays the same, and the y-coordinate flips its sign!

Example: Reflecting the point B(3, 4) across the x-axis gives the image B'(3, -4).

Reflecting Across the y-axis

When you reflect a point across the vertical y-axis, the y-axis is the mirror.

The Rule: To reflect a point \((x, y)\) across the y-axis, the image is \((-x, y)\).
This time, the y-coordinate stays the same, and the x-coordinate flips its sign!

Example: Reflecting the point C(-2, 5) across the y-axis gives the image C'(2, 5).

Memory Aid:

Here's an easy way to remember the rules:
- When you reflect in the x-axis, the x-coordinate stays the same.
- When you reflect in the y-axis, the y-coordinate stays the same.

Reflecting Across Other Horizontal or Vertical Lines

Sometimes the mirror line isn't an axis! What if you need to reflect across the line y = 2 or x = -1?

The 'Counting Squares' Method:
1. Pick a point on your object.
2. Count the number of squares from that point directly to the line of reflection.
3. Count the same number of squares on the other side of the line.
4. That's where your image point goes! Repeat for all corners of the shape.

Key Takeaway for Reflection

Reflection is a flip across a line. For reflections in the axes, just flip the sign of the other coordinate. For other lines, count the squares!

Transformation 3: Rotation (The Turn)

A rotation is a "turn". A shape is turned around a fixed point, called the centre of rotation. For our lessons, the centre of rotation is often the origin (0, 0).

To describe a rotation, you need three things:
1. The centre of rotation (e.g., the origin).
2. The angle of rotation (e.g., \(90^\circ, 180^\circ, 270^\circ\)).
3. The direction of rotation (clockwise or anti-clockwise).

Did you know?

In mathematics, anti-clockwise is considered the standard positive direction for rotation!

The Coordinate Rules for Rotation about the Origin (0,0)

For any point \((x, y)\):

Rotation of 90° anti-clockwise: The new point is \((-y, x)\).
(Trick: Swap x and y, then change the sign of the new first number)
Rotation of 180° (either direction): The new point is \((-x, -y)\).
(Trick: Just flip the signs of both x and y)
Rotation of 270° anti-clockwise: The new point is \((y, -x)\).
(Trick: Swap x and y, then change the sign of the new second number)

Quick Note: A 90° clockwise turn is the same as a 270° anti-clockwise turn. And a 270° clockwise turn is the same as a 90° anti-clockwise turn!

Step-by-Step Example:

Let's rotate the point D(4, 2) by 90° anti-clockwise about the origin.

1. Original point: \((x, y) = (4, 2)\).
2. The rule for 90° anti-clockwise is: \((x, y) \to (-y, x)\).
3. Apply the rule: The new x will be -y, so it's -2. The new y will be x, so it's 4.
4. The image point D' is at (-2, 4).

Key Takeaway for Rotation

Rotation is a turn around a point. Memorise the three key rules for 90°, 180°, and 270° anti-clockwise rotations around the origin.

Transformation 4: Enlargement and Reduction (Dilation / Contraction)

An enlargement or reduction changes the size of a shape by multiplying all side lengths by a constant scale factor \(k\), with respect to a centre of enlargement.

- If \(k > 1\), the shape gets larger (enlargement).
- If \(0 < k < 1\), the shape gets smaller (reduction).
- When the centre is the origin \((0,0)\), every coordinate \((x, y)\) transforms to \((kx, ky)\).

Example: Enlarging point P(2, 3) by a scale factor of 3 about the origin gives P'(6, 9).

Invariant Properties under Transformations

An invariant property is a feature of the shape that does not change after the transformation:

- Translation: Side lengths, angle sizes, overall shape/area, and orientation are all invariant.
- Reflection: Side lengths, angle sizes, and overall shape/area are invariant, but orientation is reversed (flipped).
- Rotation: Side lengths, angle sizes, and overall shape/area are invariant, but orientation changes direction.
- Enlargement / Reduction: Angle sizes and overall geometric shape are invariant, but side lengths, area, and coordinates change proportionally.

Part 3: Tessellations - Patterns that Fit Together

A tessellation is a pattern made of one or more shapes that fit together perfectly, without any gaps or overlaps. It's like tiling a floor!

Real-world examples: A honeycomb made by bees, a brick wall, bathroom floor tiles.

Why Do Some Shapes Tessellate?

The secret is in the corners! For shapes to tessellate, the interior angles meeting at every vertex must add up to exactly 360 degrees (\(360^\circ\)).

Which Regular Polygons Tessellate?

Only three regular polygons can tessellate all by themselves:

- Equilateral Triangles: Each angle is 60°. Six of them meet at a point (\(6 \times 60^\circ = 360^\circ\)).
- Squares: Each angle is 90°. Four of them meet at a point (\(4 \times 90^\circ = 360^\circ\)).
- Regular Hexagons: Each angle is 120°. Three of them meet at a point (\(3 \times 120^\circ = 360^\circ\)).

A regular pentagon cannot tessellate alone because its interior angle is 108°, and 108° cannot be multiplied by any whole number to equal 360°.

Did you know?

All triangles and all quadrilaterals (even irregular ones!) can tessellate the plane:
- For any triangle with angles \(\alpha, \beta, \gamma\), six triangles can meet at a vertex (two of each angle) using \(180^\circ\) rotations: \(2(\alpha + \beta + \gamma) = 2(180^\circ) = 360^\circ\).
- For any quadrilateral with angles \(\alpha, \beta, \gamma, \delta\), four copies meeting at a vertex contain one of each angle: \(\alpha + \beta + \gamma + \delta = 360^\circ\).

Key Takeaway for Tessellations

Tessellation means tiling with shapes that have no gaps or overlaps. The sum of angles meeting around every vertex must equal exactly \(360^\circ\).

Chapter Summary

Here is a final recap of everything you have learned about symmetry and transformations.

Quick Review Box

- Symmetry: A shape has balance. It can be reflectional (line symmetry) or rotational (looks identical after a partial turn).

- Translation (Slide): Moves a shape without changing size, angle, or orientation: \((x, y) \to (x+a, y+b)\).

- Reflection (Flip): Flips a shape across a line (reverses orientation):
- Across x-axis: \((x, y) \to (x, -y)\)
- Across y-axis: \((x, y) \to (-x, y)\)

- Rotation (Turn): Turns a shape around the origin \((0,0)\):
- 90° anti-clockwise: \((x, y) \to (-y, x)\)
- 180°: \((x, y) \to (-x, -y)\)
- 270° anti-clockwise: \((x, y) \to (y, -x)\)

- Enlargement / Reduction: Multiplies coordinates by scale factor \(k\): \((x, y) \to (kx, ky)\). Preserves angle sizes and shape, but changes side lengths.

- Tessellation (Tile): A repeating pattern with no gaps or overlaps, requiring angles at each vertex to sum to \(360^\circ\).

Keep practising these moves, and you'll find them in art, design, and nature all around you. Great job!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is reflectional symmetry and what is the line of symmetry?

A shape has reflectional symmetry if a straight line (the axis or line of symmetry) divides it into two halves that are mirror images of each other.

What is rotational symmetry and how is its order determined?

A shape has rotational symmetry if it looks identical to its original position after being rotated around a central point by less than a full turn (\(360^\circ\)); its order of rotational symmetry is the number of times it matches during a full \(360^\circ\) turn.

What is a translation and how do coordinates change when translating a point?

A translation is a slide of every point of a shape by the same distance and direction; a point \((x, y)\) shifted \(a\) units horizontally and \(b\) units vertically becomes \((x+a, y+b)\).

What are the coordinate rules for reflecting a point \((x, y)\) across the \(x\)-axis and the \(y\)-axis?

Reflecting across the \(x\)-axis produces \((x, -y)\), while reflecting across the \(y\)-axis produces \((-x, y)\).

How do coordinates change when a point \((x, y)\) is rotated anti-clockwise about the origin by \(90^\circ\), \(180^\circ\), and \(270^\circ\)?

About the origin, \(90^\circ\) anti-clockwise gives \((-y, x)\), \(180^\circ\) gives \((-x, -y)\), and \(270^\circ\) anti-clockwise gives \((y, -x)\).

How does an enlargement or reduction with scale factor \(k\) about the origin transform the coordinates and size of a shape?

Every coordinate \((x, y)\) becomes \((kx, ky)\), where \(k > 1\) enlarges the shape and \(0 < k < 1\) reduces it while preserving all interior angles.

Which geometric properties remain invariant under translations, reflections, rotations, and enlargements?

Translations, reflections, and rotations preserve side lengths, angles, and area (congruence); enlargements/reductions preserve angle sizes and shape proportions (similarity) but alter side lengths and area.

What geometric condition is required for shapes to form a tessellation?

The sum of the interior angles of all shapes meeting at each shared vertex must equal exactly \(360^\circ\).

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