Transformation Studio 🔄

Master reflections, rotations and translations!

🔄 What is a Transformation?

A transformation moves or flips a shape. The original shape is called the object. The result is the image.

All three transformations are isometric — they preserve size and shape. The image is congruent (identical) to the object.

There are three types we study at Stage 6:

Translation — slide in a direction
Reflection — flip over a mirror line
Rotation — turn about a centre point

➡️ Translation

A translation slides a shape without turning or flipping it. We describe it using a column vector:

(x) = right (+) or left (−)

(y) = up (+) or down (−)

Example: Vector (3, −2) means move 3 right and 2 down.

Key fact: In a translation, every point moves the same distance in the same direction. The shape does NOT rotate or reflect.

🪞 Reflection

A reflection flips a shape over a mirror line. Each point in the image is the same distance from the mirror line as the corresponding point in the object, but on the opposite side.

Mirror Line	What Happens
Vertical line (e.g., x = 3)	x-coordinates change, y stays the same
Horizontal line (e.g., y = 2)	y-coordinates change, x stays the same
Diagonal line y = x	x and y coordinates swap
Diagonal line y = −x	coordinates swap AND both change sign

Tip: To reflect, count how many squares each vertex is from the mirror line, then count the same distance on the other side.

🔃 Rotation

A rotation turns a shape about a centre of rotation. You need to know:

The centre of rotation (x, y)
The angle (90°, 180°, 270°)
The direction (clockwise or anticlockwise)

Shortcuts for rotating about the origin (0,0):

90° clockwise: (x, y) → (y, −x)

90° anticlockwise: (x, y) → (−y, x)

180°: (x, y) → (−x, −y)

Note: 90° clockwise = 270° anticlockwise. 270° clockwise = 90° anticlockwise.

⭐ Golden Rules

      Translation: describe using a column vector (right/left, up/down)
Reflection: every point stays the same distance from the mirror line
Rotation: state centre, angle AND direction
All transformations preserve size and shape (isometric)
180° rotation = same result clockwise or anticlockwise

    

✏️ Worked Examples

Example 1: Translation

Q: Triangle with vertices A(1,2), B(3,2), C(2,4) is translated by vector (4, −1). Find the new vertices.

Add the vector to each coordinate:

A(1,2) → (1+4, 2+(−1)) = A'(5, 1)

B(3,2) → (3+4, 2−1) = B'(7, 1)

C(2,4) → (2+4, 4−1) = C'(6, 3)

Example 2: Reflection in y-axis (x = 0)

Q: Reflect point P(3, 5) in the y-axis.

For reflection in the y-axis: x changes sign, y stays the same.

P(3, 5) → P'(−3, 5)

Example 3: Reflection in y = x

Q: Reflect point Q(4, 1) in the line y = x.

For reflection in y = x: swap the x and y coordinates.

Q(4, 1) → Q'(1, 4)

Example 4: Rotation 90° Clockwise about Origin

Q: Rotate R(2, 3) by 90° clockwise about the origin.

Rule: 90° clockwise: (x, y) → (y, −x)

R(2, 3) → R'(3, −2)

Example 5: Describing a Transformation

Q: Shape A has vertices (1,1),(3,1),(2,3). Shape B has vertices (−1,1),(−3,1),(−2,3). Describe the transformation.

The y-coordinates are the same. The x-coordinates have changed sign. This is a reflection in the y-axis.

🔬 Transformation Studio

Choose a transformation to see it in action on a triangle:

Click a transformation to see it drawn!

🎯 Drag 1: Translation Vectors

Match each movement description to the correct column vector.

(3, 2)

(−4, 1)

(0, −3)

(−2, −2)

(5, 0)

1. Move 3 right, 2 up

Drop here

2. Move 4 left, 1 up

Drop here

3. Move 3 down (no horizontal movement)

Drop here

4. Move 2 left, 2 down

Drop here

5. Move 5 right (no vertical movement)

Drop here

🎯 Drag 2: Translate the Point

Point A is translated. Drag the correct image coordinates.

(7, 5)

(1, 4)

(−1, 2)

(4, 1)

(3, 6)

1. A(3, 3) translated by (4, 2) → A' =

Drop here

2. A(5, 2) translated by (−4, 2) → A' =

Drop here

3. A(2, 4) translated by (−3, −2) → A' =

Drop here

4. A(6, 3) translated by (−2, −2) → A' =

Drop here

5. A(1, 4) translated by (2, 2) → A' =

Drop here

🎯 Drag 3: Reflect in the y-axis

Reflect each point in the y-axis (x changes sign, y stays the same).

(−3, 4)

(2, −1)

(−5, 0)

(4, 3)

(−1, −2)

1. Reflect (3, 4) in y-axis →

Drop here

2. Reflect (−2, −1) in y-axis →

Drop here

3. Reflect (5, 0) in y-axis →

Drop here

4. Reflect (−4, 3) in y-axis →

Drop here

5. Reflect (1, −2) in y-axis →

Drop here

🎯 Drag 4: Rotate 90° Clockwise about Origin

Apply the rule: 90° clockwise → (x, y) becomes (y, −x)

(3, −1)

(2, 4)

(0, −3)

(2, −5)

(4, −2)

1. Rotate (1, 3) 90° CW about origin →

Drop here

2. Rotate (−4, 2) 90° CW about origin →

Drop here

3. Rotate (−3, 0) 90° CW about origin →

Drop here

4. Rotate (5, 2) 90° CW about origin →

Drop here

5. Rotate (2, 4) 90° CW about origin →

Drop here

🎯 Drag 5: 180° Rotation

Apply the rule: 180° rotation about origin → (x, y) becomes (−x, −y)

(−3, −4)

(2, −1)

(0, 5)

(1, −3)

(−4, 2)

1. Rotate (3, 4) 180° about origin →

Drop here

2. Rotate (−2, 1) 180° about origin →

Drop here

3. Rotate (0, −5) 180° about origin →

Drop here

4. Rotate (−1, 3) 180° about origin →

Drop here

5. Rotate (4, −2) 180° about origin →

Drop here

🎯 Drag 6: Identify the Transformation

What transformation maps the object to its image? Drag the correct description.

Translation (4, 0)

Reflection in x-axis

Rotation 180° about origin

Reflection in y-axis

Translation (−2, 3)

1. A(1,2)→A'(5,2), B(3,2)→B'(7,2), C(2,4)→C'(6,4)

Drop here

2. A(2,3)→A'(2,−3), B(4,1)→B'(4,−1), C(3,5)→C'(3,−5)

Drop here

3. A(2,3)→A'(−2,−3), B(4,1)→B'(−4,−1), C(1,5)→C'(−1,−5)

Drop here

4. A(3,2)→A'(−3,2), B(5,4)→B'(−5,4), C(4,1)→C'(−4,1)

Drop here

5. A(4,1)→A'(2,4), B(6,2)→B'(4,5), C(5,4)→C'(3,7)

Drop here

📝 Practice Questions

1. What vector describes moving 5 right and 3 down?

(5, −3)

2. Point P(4, 6) is translated by (−3, 2). Find P'.

P'(1, 8)

3. Reflect Q(5, 3) in the y-axis.

Q'(−5, 3)

4. Reflect R(2, −4) in the x-axis.

R'(2, 4) — y changes sign, x stays

5. Rotate S(3, 1) by 90° clockwise about the origin.

S'(1, −3) — rule: (x,y)→(y,−x)

6. Rotate T(−2, 4) by 180° about the origin.

T'(2, −4) — rule: (x,y)→(−x,−y)

7. Describe the transformation: A(2,3)→A'(5,3), B(4,3)→B'(7,3).

Translation by vector (3, 0) — 3 right, no vertical movement

8. Reflect U(3, 4) in the line y = x.

U'(4, 3) — swap x and y coordinates

9. What is 90° anticlockwise equivalent to in clockwise degrees?

270° clockwise

10. Rotate V(1, 5) by 90° anticlockwise about the origin.

V'(−5, 1) — rule: (x,y)→(−y,x)

11. Triangle has vertices (2,1),(4,1),(3,3). It is translated by (0,−4). Write the new vertices.

(2,−3),(4,−3),(3,−1)

12. What transformation maps (3,2) to (−3,−2)?

Rotation of 180° about the origin

13. What transformation maps (4,2) to (2,4)?

Reflection in the line y = x (coordinates swap)

14. Does a translation change the orientation of a shape?

No — translation slides the shape without rotating or flipping it. All three transformations (translation, reflection, rotation) preserve the size and shape.

15. Reflect W(−3, 5) in the x-axis.

W'(−3, −5)

16. A point is rotated 270° clockwise about the origin. What is the equivalent anticlockwise rotation?

90° anticlockwise

17. Triangle vertices: (1,2),(3,2),(2,5). After a reflection in x = 0 (y-axis), what are the new coordinates?

(−1,2),(−3,2),(−2,5) — all x-coordinates change sign

18. What is the centre of rotation if a shape stays in the same place when rotated 360°?

Any point — a 360° rotation returns every shape to its original position regardless of centre.

19. Translate (−2, 3) by vector (5, −1).

(3, 2)

20. What transformation maps (2,5) to (5,2) to (−5,−2) in two steps?

Step 1: Reflection in y=x → (5,2). Step 2: Rotation 180° → (−5,−2).

🏆 Challenge Problems

1. Triangle T has vertices A(1,2), B(4,2), C(4,5). T is translated so that A maps to A'(3,−1). Write the translation vector and find B' and C'.

Vector: (2,−3). B'(6,−1), C'(6,2)

2. Shape A is reflected in the line x = 2. A vertex at (5, 3) maps to where?

Distance from x=2 is 5−2=3. Image is 3 to the left: (2−3, 3) = (−1, 3)

3. Point P(3,1) is rotated 90° anticlockwise about the point (1,1). Find P'.

Translate to origin: (3−1,1−1)=(2,0). Rotate 90° CCW: (−0,2)=(0,2). Translate back: (0+1,2+1)=(1,3). So P'(1,3).

4. Describe fully the single transformation that maps: A(−3,4)→A'(4,3), B(−1,2)→B'(2,1).

Reflection in the line y = −x (coordinates swap and both change sign: (x,y)→(−y,−x))

5. Shape B is translated by (3, 2) to get shape C. Shape C is then reflected in the y-axis to get shape D. If one vertex of B is at (1, 1), find the corresponding vertex of D.

B(1,1) → translate → C(4,3) → reflect in y-axis → D(−4,3)

6. A rotation maps (4,0) to (0,4). What rotation is this?

90° anticlockwise about the origin. Check: (x,y)→(−y,x): (4,0)→(0,4) ✅

7. How many times does an equilateral triangle map onto itself during a 360° rotation about its centre?

3 times — at 120°, 240° and 360°. This is its order of rotational symmetry.

8. Triangle with vertices (2,1),(5,1),(5,4) is rotated 90° clockwise about (2,1). Find the image vertices.

Centre (2,1). Relative coords: (0,0),(3,0),(3,3). After 90°CW: (0,0),(0,−3),(3,−3). Back: (2,1),(2,−2),(5,−2).

9. Is it possible to describe a reflection followed by another reflection as a single translation or rotation? Give an example.

Yes! Two reflections in parallel lines = a translation. Two reflections in intersecting lines = a rotation. E.g., reflect in x=0 then x=3 → translation by (6,0).

10. A shape is reflected in y = x, then rotated 90° clockwise about the origin. What single transformation is equivalent?

Reflect in y=x: (x,y)→(y,x). Then rotate 90°CW: (y,x)→(x,−y). Combined: (x,y)→(x,−y). This is a reflection in the x-axis!