Mathematics · Simulation and virtual lab Free
Transformations: translation, reflection, rotation and enlargement
Move a triangle with translations, reflections, rotations and enlargements, combine two of them, and solve find-the-transformation challenges.
- You will learn to
- Find the image of a shape under translation, reflection, rotation and enlargement.
- Describe a transformation fully and combine two transformations into one.
- Relate the scale factor of an enlargement to the change in area.
- You will measure
- Coordinates of A′, B′ and C′; the matrix; the area ratio for each transformation.
- Time needed
- About 30–40 min
- Equipment
- Nothing extra. Real version: squared paper, tracing paper, ruler and protractor.
- Price
- Free: no payment, no account.
Virtual lab
Follow the steps with the simulation above (go back to the simulation ↑). Each time you tap Record, a row is added to the table and a point to the graph.
Aim
To find the image of a shape under translations, reflections, rotations and enlargements, to describe a transformation fully, to combine two transformations, and to relate the scale factor of an enlargement to the change in area.
What to do
- Choose Translation. Move the triangle by (3, −5). Compare the coordinates of A and A′. Record.
- Choose Reflection and try each mirror line. Write a rule for reflecting in the y-axis and in y = x.
- Choose Rotation: 90°, 180° and −90° about O, then about P(2, 1). Record each image.
- Choose Enlargement with k = 2, 3, ½ and −1. Record each one and look at the area ratio graph.
- Choose Two in a row: reflect in the y-axis, then rotate 90° about O. Which single transformation does the same?
- Tap Challenge: find it. Find the transformation that maps ABC onto A″B″C″, then check.
Results
Questions
What are the coordinates of the image of (1, 3) after translation by the vector (−4, 2)?
(1 − 4, 3 + 2) = (−3, 5).
Reflect the point (3, 1) in the line y = x.
The coordinates swap: (1, 3).
Rotate (3, 1) through 90° anticlockwise about the origin.
(x, y) → (−y, x), so the image is (−1, 3).
A triangle of area 2 cm² is enlarged by scale factor 3. What is the area of the image?
Area × k² = 2 × 9 = 18 cm².
What information is needed to describe a rotation fully?
The centre of rotation, the angle and the direction (clockwise or anticlockwise).
Which transformations are isometries?
Translation, reflection and rotation: the image is congruent to the object (same lengths, angles and area).
For teachers: expected results and how the model works
Object A(1, 1), B(3, 1), C(1, 3), area 2. Grid from −10 to 10. Centres: O(0, 0) and P(2, 1). Mirror lines: x-axis, y-axis, y = x, y = −x, x = −1, y = −1. Every transformation is an affine map p → Mp + v; the side panel shows M, and “Two in a row” combines two maps and names the single equivalent transformation (translation, reflection, rotation with its centre, enlargement with its centre, or a glide reflection). Example: reflection in the y-axis then rotation of +90° about O gives a reflection in y = −x; rotation of 90° then reflection in the x-axis gives a reflection in y = −x as well; two reflections in parallel lines give a translation. Challenges choose a random transformation that the controls can make exactly; some images partly leave the grid. Area ratio = |det M| = k² for enlargements.
Exam practice Exam set
National-exam style questions on this simulation (16 marks). Try each one on paper first, then open the marking scheme.
Triangle ABC has A(1, 1), B(3, 1), C(1, 3). Find the image of ABC after (a) reflection in the x-axis, (b) rotation of 180° about the origin, (c) enlargement with centre O and scale factor −2. (6 marks)
Marking scheme: (a) (1, −1), (3, −1), (1, −3) (2). (b) (−1, −1), (−3, −1), (−1, −3) (2). (c) (−2, −2), (−6, −2), (−2, −6) (2).
Describe fully the single transformation equivalent to a reflection in the y-axis followed by a rotation of 90° anticlockwise about O. (3 marks)
Marking scheme: (x, y) → (−x, y) → (−y, −x) (1); this is a reflection (1) in the line y = −x (1).
Write down the matrix of (a) a reflection in the line y = x, (b) an enlargement with centre O and scale factor 3. Use it to find the image of (2, −1) in each case. (4 marks)
Marking scheme: (a) (0 1; 1 0) (1); image (−1, 2) (1). (b) (3 0; 0 3) (1); image (6, −3) (1).
A shape is enlarged with scale factor ½ about centre P(2, 1). Find the image of the point (6, 5). (3 marks)
Marking scheme: Vector from P = (4, 4) (1); × ½ = (2, 2) (1); image = (2 + 2, 1 + 2) = (4, 3) (1).
Donat Sciences and MathsMathematics · Virtual lab worksheet
School: ........................................ Name: ................................ Class: ........ Date: ............
Transformations: translation, reflection, rotation and enlargement
Aim: To find the image of a shape under translations, reflections, rotations and enlargements, to describe a transformation fully, to combine two transformations, and to relate the scale factor of an enlargement to the change in area.
Apparatus (real laboratory): squared paper, tracing paper, ruler and protractor.
What to do (with the simulation on www.donatstem.co.rw)
- Choose Translation. Move the triangle by (3, −5). Compare the coordinates of A and A′. Record.
- Choose Reflection and try each mirror line. Write a rule for reflecting in the y-axis and in y = x.
- Choose Rotation: 90°, 180° and −90° about O, then about P(2, 1). Record each image.
- Choose Enlargement with k = 2, 3, ½ and −1. Record each one and look at the area ratio graph.
- Choose Two in a row: reflect in the y-axis, then rotate 90° about O. Which single transformation does the same?
- Tap Challenge: find it. Find the transformation that maps ABC onto A″B″C″, then check.
Results
| # | Transformation | A′ | B′ | C′ | Area ratio |
|---|
| 1 | | | | | |
| 2 | | | | | |
| 3 | | | | | |
| 4 | | | | | |
| 5 | | | | | |
| 6 | | | | | |
| 7 | | | | | |
| 8 | | | | | |
| 9 | | | | | |
| 10 | | | | | |
Graph
Plot the area ratio against the scale factor k for your enlargements. Draw the curve y = k².
Questions
- What are the coordinates of the image of (1, 3) after translation by the vector (−4, 2)? [2]
- Reflect the point (3, 1) in the line y = x. [2]
- Rotate (3, 1) through 90° anticlockwise about the origin. [2]
- A triangle of area 2 cm² is enlarged by scale factor 3. What is the area of the image? [2]
- What information is needed to describe a rotation fully? [2]
- Which transformations are isometries? [2]
- Triangle ABC has A(1, 1), B(3, 1), C(1, 3). Find the image of ABC after (a) reflection in the x-axis, (b) rotation of 180° about the origin, (c) enlargement with centre O and scale factor −2. [6]
- Describe fully the single transformation equivalent to a reflection in the y-axis followed by a rotation of 90° anticlockwise about O. [3]
- Write down the matrix of (a) a reflection in the line y = x, (b) an enlargement with centre O and scale factor 3. Use it to find the image of (2, −1) in each case. [4]
- A shape is enlarged with scale factor ½ about centre P(2, 1). Find the image of the point (6, 5). [3]
Donat Sciences and MathsMathematics · Virtual lab worksheet
School: ........................................ Name: ................................ Class: ........ Date: ............
Marking scheme: Transformations: translation, reflection, rotation and enlargement
Expected results
Object A(1, 1), B(3, 1), C(1, 3), area 2. Grid from −10 to 10. Centres: O(0, 0) and P(2, 1). Mirror lines: x-axis, y-axis, y = x, y = −x, x = −1, y = −1. Every transformation is an affine map p → Mp + v; the side panel shows M, and “Two in a row” combines two maps and names the single equivalent transformation (translation, reflection, rotation with its centre, enlargement with its centre, or a glide reflection). Example: reflection in the y-axis then rotation of +90° about O gives a reflection in y = −x; rotation of 90° then reflection in the x-axis gives a reflection in y = −x as well; two reflections in parallel lines give a translation. Challenges choose a random transformation that the controls can make exactly; some images partly leave the grid. Area ratio = |det M| = k² for enlargements.
Answers
- What are the coordinates of the image of (1, 3) after translation by the vector (−4, 2)? [2]
(1 − 4, 3 + 2) = (−3, 5).
- Reflect the point (3, 1) in the line y = x. [2]
The coordinates swap: (1, 3).
- Rotate (3, 1) through 90° anticlockwise about the origin. [2]
(x, y) → (−y, x), so the image is (−1, 3).
- A triangle of area 2 cm² is enlarged by scale factor 3. What is the area of the image? [2]
Area × k² = 2 × 9 = 18 cm².
- What information is needed to describe a rotation fully? [2]
The centre of rotation, the angle and the direction (clockwise or anticlockwise).
- Which transformations are isometries? [2]
Translation, reflection and rotation: the image is congruent to the object (same lengths, angles and area).
- Triangle ABC has A(1, 1), B(3, 1), C(1, 3). Find the image of ABC after (a) reflection in the x-axis, (b) rotation of 180° about the origin, (c) enlargement with centre O and scale factor −2. [6]
(a) (1, −1), (3, −1), (1, −3) (2). (b) (−1, −1), (−3, −1), (−1, −3) (2). (c) (−2, −2), (−6, −2), (−2, −6) (2).
- Describe fully the single transformation equivalent to a reflection in the y-axis followed by a rotation of 90° anticlockwise about O. [3]
(x, y) → (−x, y) → (−y, −x) (1); this is a reflection (1) in the line y = −x (1).
- Write down the matrix of (a) a reflection in the line y = x, (b) an enlargement with centre O and scale factor 3. Use it to find the image of (2, −1) in each case. [4]
(a) (0 1; 1 0) (1); image (−1, 2) (1). (b) (3 0; 0 3) (1); image (6, −3) (1).
- A shape is enlarged with scale factor ½ about centre P(2, 1). Find the image of the point (6, 5). [3]
Vector from P = (4, 4) (1); × ½ = (2, 2) (1); image = (2 + 2, 1 + 2) = (4, 3) (1).