🔄 Transformations — Grade 5

Learn to reflect, translate, and rotate shapes on a coordinate grid!

1. Reflection

A reflection flips a shape over a mirror line. The image is the same distance from the mirror line as the original.

Reflect in x-axis

(x, y) → (x, −y)
y-value changes sign

Reflect in y-axis

(x, y) → (−x, y)
x-value changes sign

Reflect in y = k

y stays the same,
x reflected about x = k

Reflect (3, 4) in the x-axis: y changes sign → (3, −4)

Reflect (3, 4) in the y-axis: x changes sign → (−3, 4)

2. Translation

A translation slides a shape. We write the movement as a column vector.

The vector (a over b) means: move a in x-direction, b in y-direction.

Translate (2, 5) by vector (3, −2):

New x = 2 + 3 = 5

New y = 5 + (−2) = 3

Image = (5, 3)

3. Rotation

A rotation turns a shape around a centre of rotation. Always state the angle and direction.

90° Clockwise

(x, y) → (y, −x)

90° Anticlockwise

(x, y) → (−y, x)

180°

(x, y) → (−x, −y)

Rotate (3, 2) by 90° clockwise about origin:

(x, y) → (y, −x) = (2, −3)

4. Describing Transformations

Always give full details:

Reflection: state the mirror line (e.g. "reflection in the x-axis")
Translation: state the column vector
Rotation: state the angle, direction, and centre of rotation

5. Symmetry and Transformations

Rotational symmetry order = how many times a shape looks the same in one full 360° turn.

Square

Order 4

Equilateral Triangle

Order 3

Rectangle

Order 2

Regular Hexagon

Order 6

Circle

Order ∞ (enter 0)

Scalene Triangle

Order 1

🔄 Worked Examples

Example 1 — Reflection in x-axis

Reflect (5, −3) in the x-axis. Find new y.

(x, y) → (x, −y): new y = −(−3) = 3

Example 2 — Translation

Translate (−2, 4) by vector (6, −3). Find new x and y.

New x = −2 + 6 = 4

New y = 4 + (−3) = 1

Example 3 — Rotation 90° CW

Rotate (4, 1) by 90° clockwise about origin.

(x, y) → (y, −x) = (1, −4)

New x = 1, New y = −4

Example 4 — Rotation 180°

Rotate (−3, 5) by 180° about origin.

(x, y) → (−x, −y) = (3, −5)

New x = 3, New y = −5

Example 5 — Rotational Symmetry

What is the order of rotational symmetry of a regular hexagon?

A regular hexagon looks the same 6 times in a full turn. Order = 6

🔭 Transformation Explorer

Transform a Triangle

The pink triangle is at (1,1), (4,1), (1,4). Choose a transformation to see the green image.

Transformation

Exercise 1 — Reflection

Reflect each point. Reflect in x-axis: (x,y)→(x,−y). Reflect in y-axis: (x,y)→(−x,y).

1. Reflect (3, 4) in the x-axis. Enter new y.
2. Reflect (5, −2) in the x-axis. Enter new y.
3. Reflect (−1, 6) in the x-axis. Enter new y.
4. Reflect (4, 3) in the y-axis. Enter new x.
5. Reflect (−3, 5) in the y-axis. Enter new x.
6. Reflect (2, −4) in the y-axis. Enter new x.
7. Reflect (0, 7) in the x-axis. Enter new y.
8. Reflect (−5, −3) in the y-axis. Enter new x.
9. Reflect (6, 2) in the x-axis. Enter new y.
10. Reflect (−4, 1) in the y-axis. Enter new x.

Exercise 2 — Translation

Translate each point by the given vector.

1. Translate (2, 3) by (5, 2). Enter new x.
2. Translate (2, 3) by (5, 2). Enter new y.
3. Translate (4, −1) by (−3, 4). Enter new x.
4. Translate (4, −1) by (−3, 4). Enter new y.
5. Translate (−2, 5) by (4, −7). Enter new x.
6. Translate (−2, 5) by (4, −7). Enter new y.
7. Translate (0, 0) by (−5, 3). Enter new x.
8. Translate (0, 0) by (−5, 3). Enter new y.
9. Translate (3, −4) by (2, 6). Enter new x.
10. Translate (3, −4) by (2, 6). Enter new y.

Exercise 3 — Rotation 90° Clockwise

Rule: (x, y) → (y, −x). Rotate about origin unless stated.

1. Rotate (3, 2) by 90°CW. Enter new x.
2. Rotate (3, 2) by 90°CW. Enter new y.
3. Rotate (−1, 4) by 90°CW. Enter new x.
4. Rotate (−1, 4) by 90°CW. Enter new y.
5. Rotate (5, −3) by 90°CW. Enter new x.
6. Rotate (5, −3) by 90°CW. Enter new y.
7. Rotate (0, 4) by 90°CW. Enter new x.
8. Rotate (0, 4) by 90°CW. Enter new y.
9. Rotate (−2, −3) by 90°CW. Enter new x.
10. Rotate (−2, −3) by 90°CW. Enter new y.

Exercise 4 — Rotation 180°

Rule: (x, y) → (−x, −y). Rotate 180° about origin.

1. Rotate (4, 3) by 180°. Enter new x.
2. Rotate (4, 3) by 180°. Enter new y.
3. Rotate (−2, 5) by 180°. Enter new x.
4. Rotate (−2, 5) by 180°. Enter new y.
5. Rotate (6, −4) by 180°. Enter new x.
6. Rotate (6, −4) by 180°. Enter new y.
7. Rotate (−3, −1) by 180°. Enter new x.
8. Rotate (−3, −1) by 180°. Enter new y.
9. Rotate (0, 5) by 180°. Enter new x.
10. Rotate (0, 5) by 180°. Enter new y.

Exercise 5 — Order of Rotational Symmetry

Enter the order of rotational symmetry. Circle = 0 (infinite).

1. Square
2. Equilateral triangle
3. Rectangle (not square)
4. Regular hexagon
5. Circle (enter 0 for infinite)
6. Scalene triangle
7. Regular pentagon
8. Rhombus
9. Isosceles triangle (non-equilateral)
10. Regular octagon

Practice — 20 Questions

1. Reflect (4, 5) in x-axis. New y = ?
2. Reflect (−3, 2) in y-axis. New x = ?
3. Translate (1, 2) by (4, 3). New x = ?
4. Translate (1, 2) by (4, 3). New y = ?
5. Rotate (2, 5) 90°CW. New x = ?
6. Rotate (2, 5) 90°CW. New y = ?
7. Rotate (3, 4) 180°. New x = ?
8. Order of rotational symmetry of a square?
9. Reflect (6, −2) in x-axis. New y = ?
10. Translate (−3, 4) by (2, −6). New y = ?
11. Rotate (−4, 1) 90°CW. New x = ?
12. Rotate (0, 3) 180°. New y = ?
13. Order of rotational symmetry of rectangle?
14. Reflect (−2, −5) in y-axis. New x = ?
15. Translate (5, −1) by (−3, 4). New y = ?
16. Rotate (4, −2) 90°ACW. Rule: (x,y)→(−y,x). New x = ?
17. Order of rotational symmetry of regular hexagon?
18. Rotate (−1, 3) 180°. New x = ?
19. Reflect (7, 4) in x-axis. New y = ?
20. Order of rotational symmetry of equilateral triangle?

Challenge — 8 Hard Questions

1. A point A(3,2) is reflected in x-axis to get A', then A' is reflected in y-axis to get A''. What is the x-coordinate of A''?
2. A point A(3,2) reflected twice as above. What is the y-coordinate of A''?
3. Point P(5,3) is rotated 90°CW, then the result is rotated 90°CW again. Enter new x.
4. Point P(5,3) rotated twice 90°CW. Enter new y.
5. Triangle vertices: (1,2),(3,2),(2,4). After translation (−4,1) the new vertex that was (1,2) has x = ?
6. Shape A is at (2,3). Shape B is at (−2,3). The transformation from A to B is a reflection. The mirror line is x = ?
7. Rotating 90°CW then 90°CW again = rotating by 180°. Rotate (4,−1) by 180°. Enter new x.
8. A regular polygon has rotational symmetry order 5. How many sides does it have?