GCSE Maths: Describing and identifying single transformations (Higher)
Part of the GCSE Maths transformations and similarity chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Describing and identifying single transformations
In an examination the question often says “Describe fully the single transformation that maps shape A onto shape B.” You must (1) name the transformation and (2) give the extra detail that each type needs. Without the detail you do not get the marks. The table shows what you need.
| Transformation | You must give | Size | Facing |
|---|---|---|---|
| Reflection | the equation of the mirror line | same (congruent) | flipped |
| Rotation | angle, direction (not for 180°), centre | same (congruent) | turned |
| Translation | the column vector | same (congruent) | same way, sides parallel |
| Enlargement | scale factor and centre | changed (similar) | same way (unless the scale factor is negative) |
How to decide. Ask these questions in order. (1) Has the size changed? If yes, it is an enlargement. (2) Is the shape flipped over (a mirror image)? If yes, it is a reflection. (3) Are all the sides parallel to the sides of the original, with the shape facing the same way? If yes, it is a translation. (4) Otherwise it has turned, so it is a rotation. Then find the detail: the line, the vector, the centre and angle, or the scale factor and centre.
Invariant points. A point that does not move is called an invariant point. Every point on the mirror line of a reflection is invariant; the centre of a rotation or an enlargement is invariant; a translation has none.
Key ideas
- “Describe fully” means: name the transformation and give its detail.
- Reflection: equation of the mirror line. Rotation: angle, direction, centre. Translation: column vector. Enlargement: scale factor, centre.
- Reflections, rotations and translations give congruent images. Enlargements give similar images.
- Reflection turns the shape into a mirror image; rotation turns it; translation slides it; enlargement resizes it.
- Invariant point: a point that does not move.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. A reflection flips the shape over. All other transformations of this type keep the shape facing the same way.Compare a side: it is 3 long in A and 3 long in B.
The size is unchanged, but B is the mirror image of A: if you go round the corners in the same order, one shape is clockwise and the other is anticlockwise. So the transformation is a reflection. - Pick vertices and their images, and find the midpoints. A mirror line is exactly half-way between an object point and its image, so find the midpoint of a point and its image.(1, 3) and (1, −1): midpoint = ((1 + 1) ÷ 2, (3 + −1) ÷ 2) = (1, 1)
(3, 3) and (3, −1): midpoint = ((3 + 3) ÷ 2, (3 + −1) ÷ 2) = (3, 1) - Write the equation of the mirror line. The midpoints all have y = 1. The points are in a horizontal line, so the mirror line is the horizontal line y = 1.Mirror line: y = 1
- Write the full description. We have found every piece of information needed for this transformation.Reflection in the line y = 1
Check. Reflect (3, 3) in y = 1 and we get (3, −1), which matches (3, −1) in B. ✓
Watch out. A reflection needs the equation of the mirror line. Writing just “reflection” or “reflect in the x-axis” for a line like x = 3 is wrong.
Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.Compare a side: it is 2 long in A and 2 long in B.
The size is unchanged, B is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation. - Find the centre of rotation. The centre is the same distance from a point and from its image. The origin is the first place to try. It works if the squared distances from the origin to the point and its image match.Distance2 of (2, 2) from (0, 0) = 22 + 22 = 8
Distance2 of (−2, 2) from (0, 0) = −22 + 22 = 8
They match, and the same works for the other vertices, so the centre is (0, 0). - Find the angle and direction. Write the move from C to a vertex, and the move from C to its image. A quarter turn changes (a, b) in a special way: clockwise gives (b, −a) and anticlockwise gives (−b, a).From C to (2, 2): (2, 2) (2 right, 2 up)
From C to (−2, 2): (−2, 2) (2 left, 2 up)
This matches (a, b) → (−b, a), so the turn is 90° anticlockwise. - Write the full description. We have found every piece of information needed for this transformation.Rotation, 90° anticlockwise, centre (0, 0)
Check. Rotate (4, 2) about (0, 0) by 90° anticlockwise: we get (−2, 4), which matches the vertex (−2, 4) of B. ✓
Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. Reflections, rotations and translations never change the size of a shape. If the size changes, it must be an enlargement.Compare a side: it is 1 long in A and 3 long in B.
The sides have changed length but the shape is the same, so the transformation is an enlargement. - Find the size of the scale factor. Scale factor = length on the image ÷ matching length on the object.A side of A is 1 and the matching side of B is 3.
3 ÷ 1 = 3 - Decide whether it is positive or negative. B is the right way up and on the same side of the centre, so the scale factor is positive. It is larger than A.Scale factor = 3
- Find the centre of enlargement. Draw a straight line through a vertex of the object and the matching vertex of the image. Do the same for a second vertex. The centre is where the lines cross. (The lines are found here from their equations.)Line through (2, 1) and (4, 1): y = 1
Line through (2, 2) and (4, 4): y = x
They cross at (1, 1), so C = (1, 1). - Write the full description. We have found every piece of information needed for this transformation.Enlargement, scale factor 3, centre (1, 1)
Check. Enlarge (2, 1) from (1, 1) with scale factor 3: (1 + 3 × 1, 1 + 3 × 0) = (4, 1), which is the matching vertex of B. ✓
Watch out. An enlargement needs both the scale factor and the centre. Do not forget that a fractional scale factor still means “enlargement”.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Think about what the transformation does to each point, then find which points stay where they are.
- Say what an invariant point is. An invariant point is a point that does not move when the transformation is done.Invariant = unchanged
- Think about what a rotation does. Every point is moved (turned or stretched) about the centre, except the centre itself: it is at distance 0 from the centre, so it cannot move.The centre is (1, 2).
- Test the centre. Apply the transformation to the centre.(1, 2) → (1, 2)
- Write the answer. Only the centre is invariant (the scale factor is not 1, and the turn is not 0° or 360°).Invariant point: (1, 2) only.
Check. Test another point, (2, 3): it goes to (2, 1), which is different, so it is not invariant. ✓
Watch out. Do not confuse “invariant point” with “image”. An invariant point is its own image.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. A reflection flips the shape over. All other transformations of this type keep the shape facing the same way.Compare the areas: 4 square units for A and 4 for B.
The size is unchanged, but B is the mirror image of A: if you go round the corners in the same order, one shape is clockwise and the other is anticlockwise. So the transformation is a reflection. - Compare the coordinates of a vertex and its image. For the diagonal mirror lines y = x and y = −x the coordinates change in a special way.The coordinates have been swapped and both signs changed: (2, 1) became (−1, −2).
- Use the midpoint as a check. The mirror line must pass through the midpoint of a point and its image.Midpoint of (2, 1) and (−1, −2) = (0.5, −0.5), which lies on y = −x.
- Write the equation of the mirror line. A reflection that swaps x and y and changes both signs is a reflection in the line y = −x.Mirror line: y = −x
- Write the full description. We have found every piece of information needed for this transformation.Reflection in the line y = −x
Check. Reflect (5, 2) in y = −x and we get (−2, −5), which matches (−2, −5) in B. ✓
Watch out. A reflection needs the equation of the mirror line. Writing just “reflection” or “reflect in the x-axis” for a line like x = 3 is wrong.
Watch out
Never write just “reflection” or “enlargement”. Each type needs its detail: the mirror line, the centre and scale factor, and so on. Also do not write two transformations: the question asks for a single one. If the shape has got bigger, it is an enlargement, even if it has also moved.Exercise 21E Describing single transformations
Non-calculatorMethod. Think about which transformation flips a shape over.
- Check the size. The size is the same, so it cannot be an enlargement.Same size: reflection, rotation or translation
- Check the facing. A mirror image is a flipped shape. Only a reflection flips a shape.Reflection
Check. A rotation or translation never flips a shape. ✓
Watch out. You would still need the equation of the mirror line to describe it fully.
Method. Look for a change of size.
- Check the size. Reflections, rotations and translations never change the size of a shape.B is bigger than A, so it is not any of those three.
- Name the transformation. Same shape but a different size is an enlargement.Enlargement
Check. The angles of A and B are equal, which fits an enlargement. ✓
Watch out. You would still need the scale factor and the centre.
Method. Same size, same facing and parallel sides means the shape has only slid.
- Check the size. The size is the same.Not an enlargement
- Check the facing. The shape is not flipped, and its sides are parallel to the originals, so it has not turned either.Translation
Check. A rotation (other than 360°) changes the direction of the sides. ✓
Watch out. You would still need the column vector.
Method. A turned shape that is not flipped is a rotation.
- Check the size. Same size.Not an enlargement
- Check for a flip. The shape is not a mirror image.Not a reflection
- Check for a slide. The shape has turned, so it is not a translation.Rotation
Check. The sides of B are not parallel to the sides of A. ✓
Watch out. You would still need the angle, direction and centre.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. A reflection flips the shape over. All other transformations of this type keep the shape facing the same way.Compare the areas: 4 square units for A and 4 for B.
The size is unchanged, but B is the mirror image of A: if you go round the corners in the same order, one shape is clockwise and the other is anticlockwise. So the transformation is a reflection. - Compare the coordinates of a vertex and its image. For the diagonal mirror lines y = x and y = −x the coordinates change in a special way.The coordinates have been swapped: (2, 1) became (1, 2).
- Use the midpoint as a check. The mirror line must pass through the midpoint of a point and its image.Midpoint of (2, 1) and (1, 2) = (1.5, 1.5), which lies on y = x.
- Write the equation of the mirror line. A reflection that swaps x and y is a reflection in the line y = x.Mirror line: y = x
- Write the full description. We have found every piece of information needed for this transformation.Reflection in the line y = x
Check. Reflect (5, 2) in y = x and we get (2, 5), which matches (2, 5) in B. ✓
Watch out. A reflection needs the equation of the mirror line. Writing just “reflection” or “reflect in the x-axis” for a line like x = 3 is wrong.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. A reflection flips the shape over. All other transformations of this type keep the shape facing the same way.Compare a side: it is 3 long in A and 3 long in B.
The size is unchanged, but B is the mirror image of A: if you go round the corners in the same order, one shape is clockwise and the other is anticlockwise. So the transformation is a reflection. - Pick vertices and their images, and find the midpoints. A mirror line is exactly half-way between an object point and its image, so find the midpoint of a point and its image.(1, 1) and (9, 1): midpoint = ((1 + 9) ÷ 2, (1 + 1) ÷ 2) = (5, 1)
(4, 1) and (6, 1): midpoint = ((4 + 6) ÷ 2, (1 + 1) ÷ 2) = (5, 1) - Write the equation of the mirror line. The midpoints all have x = 5. The points are in a vertical line, so the mirror line is the vertical line x = 5.Mirror line: x = 5
- Write the full description. We have found every piece of information needed for this transformation.Reflection in the line x = 5
Check. Reflect (4, 1) in x = 5 and we get (6, 1), which matches (6, 1) in B. ✓
Watch out. A reflection needs the equation of the mirror line. Writing just “reflection” or “reflect in the x-axis” for a line like x = 3 is wrong.
Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.Compare a side: it is 3 long in A and 3 long in B.
The size is unchanged, B is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation. - Find the centre of rotation. The centre is the same distance from a point and from its image. The origin is the first place to try. It works if the squared distances from the origin to the point and its image match.Distance2 of (2, 2) from (0, 0) = 22 + 22 = 8
Distance2 of (2, −2) from (0, 0) = 22 + −22 = 8
They match, and the same works for the other vertices, so the centre is (0, 0). - Find the angle and direction. Write the move from C to a vertex, and the move from C to its image. A quarter turn changes (a, b) in a special way: clockwise gives (b, −a) and anticlockwise gives (−b, a).From C to (2, 2): (2, 2) (2 right, 2 up)
From C to (2, −2): (2, −2) (2 right, 2 down)
This matches (a, b) → (b, −a), so the turn is 90° clockwise. - Write the full description. We have found every piece of information needed for this transformation.Rotation, 90° clockwise, centre (0, 0)
Check. Rotate (4, 2) about (0, 0) by 90° clockwise: we get (2, −4), which matches the vertex (2, −4) of B. ✓
Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.
−5
Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. A translation only slides the shape. If the shape had turned, its sides would no longer be parallel to the originals.Compare a side: it is 3 long in A and 3 long in B.
The size is unchanged, B faces the same way as A, and every side is parallel to its original. So the transformation is a translation. - Pick a vertex and its image. Every point moves by the same amount, so one vertex is enough to find the move.(2, 2) → (5, −3)
- Find the move across and the move up or down. Subtract the x-coordinates, then the y-coordinates (image − object). Positive means right/up, negative means left/down.Across: 5 − 2 = 3 (3 right)
Up/down: −3 − 2 = −5 (5 down) - Write the move as a column vector. The across move goes on top and the up/down move goes underneath.Translation by 3
−5 - Write the full description. We have found every piece of information needed for this transformation.Translation by 3
−5
Check. Test another vertex: (4, 2) + 3
−5 = (7, −3), which is the matching vertex of B. ✓
Watch out. Write the vector as across on top, up/down underneath, and take image − object, not the other way round.
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. Reflections, rotations and translations never change the size of a shape. If the size changes, it must be an enlargement.Compare a side: it is 3 long in A and 6 long in B.
The sides have changed length but the shape is the same, so the transformation is an enlargement. - Find the size of the scale factor. Scale factor = length on the image ÷ matching length on the object.A side of A is 3 and the matching side of B is 6.
6 ÷ 3 = 2 - Decide whether it is positive or negative. B is the right way up and on the same side of the centre, so the scale factor is positive. It is larger than A.Scale factor = 2
- Find the centre of enlargement. Draw a straight line through a vertex of the object and the matching vertex of the image. Do the same for a second vertex. The centre is where the lines cross. (The lines are found here from their equations.)Line through (1, 1) and (2, 2): y = x
Line through (4, 1) and (8, 2): y = 0.25x
They cross at (0, 0), so C = (0, 0). - Write the full description. We have found every piece of information needed for this transformation.Enlargement, scale factor 2, centre (0, 0)
Check. Enlarge (1, 1) from (0, 0) with scale factor 2: (0 + 2 × 1, 0 + 2 × 1) = (2, 2), which is the matching vertex of B. ✓
Watch out. An enlargement needs both the scale factor and the centre. Do not forget that a fractional scale factor still means “enlargement”.
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. Reflections, rotations and translations never change the size of a shape. If the size changes, it must be an enlargement.Compare a side: it is 6 long in A and 3 long in B.
The sides have changed length but the shape is the same, so the transformation is an enlargement. - Find the size of the scale factor. Scale factor = length on the image ÷ matching length on the object.A side of A is 6 and the matching side of B is 3.
3 ÷ 6 = 12 - Decide whether it is positive or negative. B is the right way up and on the same side of the centre, so the scale factor is positive. It is smaller than A, so the scale factor is a fraction between 0 and 1.Scale factor = 12
- Find the centre of enlargement. Draw a straight line through a vertex of the object and the matching vertex of the image. Do the same for a second vertex. The centre is where the lines cross. (The lines are found here from their equations.)Line through (2, 2) and (1, 1): y = x
Line through (8, 2) and (4, 1): y = 0.25x
They cross at (0, 0), so C = (0, 0). - Write the full description. We have found every piece of information needed for this transformation.Enlargement, scale factor 12, centre (0, 0)
Check. Enlarge (2, 2) from (0, 0) with scale factor 12: (0 + 12 × 2, 0 + 12 × 2) = (1, 1), which is the matching vertex of B. ✓
Watch out. An enlargement needs both the scale factor and the centre. Do not forget that a fractional scale factor still means “enlargement”.
Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.Compare a side: it is 3 long in A and 3 long in B.
The size is unchanged, B is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation. - Find the centre of rotation using perpendicular bisectors. The centre is the same distance from a point and from its image. So it lies on the perpendicular bisector of the line joining them (the line at right angles through the midpoint). The centre is where two such bisectors cross.For (2, 2) → (0, 2): the points (x, y) that are equally far from both satisfy x = 1.
For (5, 2) → (0, 5): the points equally far from both satisfy 5x − 3y = 2.
Solving the two equations together gives x = 1, y = 1.
Check in both: 1(1) + 0(1) = 1 ✓; 5(1) + −3(1) = 2 ✓.
The centre is (1, 1). - Find the angle and direction. Write the move from C to a vertex, and the move from C to its image. A quarter turn changes (a, b) in a special way: clockwise gives (b, −a) and anticlockwise gives (−b, a).From C to (2, 2): (1, 1) (1 right, 1 up)
From C to (0, 2): (−1, 1) (1 left, 1 up)
This matches (a, b) → (−b, a), so the turn is 90° anticlockwise. - Write the full description. We have found every piece of information needed for this transformation.Rotation, 90° anticlockwise, centre (1, 1)
Check. Rotate (5, 2) about (1, 1) by 90° anticlockwise: we get (0, 5), which matches the vertex (0, 5) of B. ✓
Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. A reflection flips the shape over. All other transformations of this type keep the shape facing the same way.Compare a side: it is 3 long in A and 3 long in B.
The size is unchanged, but B is the mirror image of A: if you go round the corners in the same order, one shape is clockwise and the other is anticlockwise. So the transformation is a reflection. - Compare the coordinates of a vertex and its image. For the diagonal mirror lines y = x and y = −x the coordinates change in a special way.The coordinates have been swapped and both signs changed: (1, 2) became (−2, −1).
- Use the midpoint as a check. The mirror line must pass through the midpoint of a point and its image.Midpoint of (1, 2) and (−2, −1) = (−0.5, 0.5), which lies on y = −x.
- Write the equation of the mirror line. A reflection that swaps x and y and changes both signs is a reflection in the line y = −x.Mirror line: y = −x
- Write the full description. We have found every piece of information needed for this transformation.Reflection in the line y = −x
Check. Reflect (3, 2) in y = −x and we get (−2, −3), which matches (−2, −3) in B. ✓
Watch out. A reflection needs the equation of the mirror line. Writing just “reflection” or “reflect in the x-axis” for a line like x = 3 is wrong.
Method. Congruent means exactly the same size and shape.
- Think about what a reflection does. It flips the shape but does not stretch or shrink it.Same size, same shape
- Decide. Congruent
Check. You could fold along the mirror line and the shapes would match. ✓
Watch out. Congruent shapes can be flipped over: they are still congruent.
Method. Congruent means exactly the same size and shape.
- Think about what a rotation does. It turns the shape but does not change any length.Same size, same shape
- Decide. Congruent
Check. Trace the shape and turn the tracing paper: it fits exactly. ✓
Watch out. Turning does not change size.
Method. Congruent means exactly the same size and shape.
- Think about what a translation does. It slides the shape without changing it.Same size, same shape
- Decide. Congruent
Check. Slide the tracing paper over: it fits. ✓
Watch out. Moving a shape does not change it.
Method. Compare the sizes.
- Think about what this enlargement does. Every length becomes 3 times as long, but the angles do not change.Same shape, different size
- Decide. Same shape and different size means similar.Similar, not congruent
Check. A side of 2 cm becomes 6 cm, so the shapes are not the same size. ✓
Watch out. An enlargement with scale factor 1 would be congruent, but 3 is not 1.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Think about what the transformation does to each point, then find which points stay where they are.
- Say what an invariant point is. An invariant point is a point that does not move when the transformation is done. Its image is the point itself.Invariant = unchanged
- Think about what a reflection does. A point on the mirror line is already on the mirror, so it stays where it is. A point not on the line moves to the other side.The mirror line is x = 2.
- Test one point on the line. Reflect it and see if it moves.(2, 3) is on the line, and its image is (2, 3), which is the same point.
- Write the answer. All points on the mirror line are invariant, and no others.Invariant points: every point on the mirror line x = 2.
Check. Test a point off the line, such as (3, 5): it goes to (1, 5), so it is not invariant. ✓
Watch out. Do not confuse “invariant point” with “image”. An invariant point is its own image.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Think about what the transformation does to each point, then find which points stay where they are.
- Say what an invariant point is. An invariant point is a point that does not move when the transformation is done.Invariant = unchanged
- Think about what a rotation does. Every point is moved (turned or stretched) about the centre, except the centre itself: it is at distance 0 from the centre, so it cannot move.The centre is (1, 1).
- Test the centre. Apply the transformation to the centre.(1, 1) → (1, 1)
- Write the answer. Only the centre is invariant (the scale factor is not 1, and the turn is not 0° or 360°).Invariant point: (1, 1) only.
Check. Test another point, (2, 2): it goes to (2, 0), which is different, so it is not invariant. ✓
Watch out. Do not confuse “invariant point” with “image”. An invariant point is its own image.
2
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Think about what the transformation does to each point, then find which points stay where they are.
- Say what an invariant point is. An invariant point is a point that does not move when the transformation is done.Invariant = unchanged
- Think about what a translation does. A translation moves every point by the same vector, and the vector is not zero.Every point moves by 3
2. - Test a point. See if its image is the same as the point.(0, 0) → (3, 2), which is different.
- Write the answer. No point stays where it is.There are no invariant points.
Check. For a point to stay put we would need to add 0 to both coordinates, but we add 3 and 2. ✓
Watch out. Do not confuse “invariant point” with “image”. An invariant point is its own image.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Think about what the transformation does to each point, then find which points stay where they are.
- Say what an invariant point is. An invariant point is a point that does not move when the transformation is done.Invariant = unchanged
- Think about what a enlargement does. Every point is moved (turned or stretched) about the centre, except the centre itself: it is at distance 0 from the centre, so it cannot move.The centre is (0, 0).
- Test the centre. Apply the transformation to the centre.(0, 0) → (0, 0)
- Write the answer. Only the centre is invariant (the scale factor is not 1, and the turn is not 0° or 360°).Invariant point: (0, 0) only.
Check. Test another point, (1, 1): it goes to (3, 3), which is different, so it is not invariant. ✓
Watch out. Do not confuse “invariant point” with “image”. An invariant point is its own image.
Method. Think about how to undo a reflection.
- Think about one point. A point P is reflected to P′. P′ is the same distance from the mirror on the other side.P and P′ are mirror images of each other.
- Undo it. Reflecting P′ in the same line puts it back at P, because the mirror line is half-way between them.Same mirror line
- Write the description. Reflection in the line x = 6
Check. We saw in 21.1 that two reflections in the same line cancel. ✓
Watch out. Reflections are their own inverse.
Method. To undo an enlargement, shrink by the same amount about the same centre.
- Think about the scale factor. A went to B by multiplying all distances from the centre by 3. To get back, divide them by 3, which is multiplying by 13.New scale factor = 13
- Think about the centre. The centre does not move, so it is the same centre.Centre (2, 1)
- Write the description. Enlargement, scale factor 13, centre (2, 1)
Check. 3 × 13 = 1, so the two enlargements cancel. ✓
Watch out. The inverse of scale factor k is 1k, not −k.
Method. The mirror line must be half-way between a point and its image.
- Test Ali’s line. The y-axis is x = 0. The image of (3, 2) in the y-axis would be (−3, 2).(3, 2) → (−3, 2), but B has (7, 2). Ali is wrong.
- Find the real mirror line. Find the midpoint of (3, 2) and (7, 2).((3 + 7) ÷ 2, (2 + 2) ÷ 2) = (5, 2)
- Write the equation. The points are level, so the mirror line is vertical.x = 5
Check. (3, 2) is 2 to the left of x = 5 and (7, 2) is 2 to the right. ✓
Watch out. Find the midpoint of a point and its image to get the mirror line.
Method. Find the image of (4, −3) under each transformation.
- Rotation of 180° about O. (a, b) → (−a, −b).(4, −3) → (−4, 3)
- Enlargement, scale factor −1, centre O. Multiply both coordinates by −1.(4 × (−1), −3 × (−1)) = (−4, 3)
- Compare. The images are the same.Both give (−4, 3).
Check. Both send every point straight through the origin to the opposite side, the same distance away. ✓
Watch out. So a 180° rotation may legitimately be described as either; in an exam, use a rotation unless the question says otherwise.
Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.Compare a side: it is 3 long in A and 3 long in B.
The size is unchanged, B is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation. - Find the centre of rotation using perpendicular bisectors. The centre is the same distance from a point and from its image. So it lies on the perpendicular bisector of the line joining them (the line at right angles through the midpoint). The centre is where two such bisectors cross.For (1, 1) → (2, 6): the points (x, y) that are equally far from both satisfy x + 5y = 19.
For (4, 1) → (2, 3): the points equally far from both satisfy x − y = 1.
Solving the two equations together gives x = 4, y = 3.
Check in both: 1(4) + 5(3) = 19 ✓; 1(4) + −1(3) = 1 ✓.
The centre is (4, 3). - Find the angle and direction. Write the move from C to a vertex, and the move from C to its image. A quarter turn changes (a, b) in a special way: clockwise gives (b, −a) and anticlockwise gives (−b, a).From C to (1, 1): (−3, −2) (3 left, 2 down)
From C to (2, 6): (−2, 3) (2 left, 3 up)
This matches (a, b) → (b, −a), so the turn is 90° clockwise. - Write the full description. We have found every piece of information needed for this transformation.Rotation, 90° clockwise, centre (4, 3)
Check. Rotate (4, 1) about (4, 3) by 90° clockwise: we get (2, 3), which matches the vertex (2, 3) of B. ✓
Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.
Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.Compare a side: it is 3 long in A and 3 long in B.
The size is unchanged, B is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation. - Find the centre of rotation. In a half turn, the centre is exactly half-way between a point and its image. Find the midpoint for two different vertices: they should be the same point.(−4, 1) and (2, 5): midpoint (−1, 3)
(−1, 1) and (−1, 5): midpoint (−1, 3)
Both midpoints are the same, so the centre is (−1, 3). - Find the angle and direction. From C the moves to the vertex and to its image point in exactly opposite directions, so it is a half turn. For a half turn the direction does not matter.From C to (−4, 1): (−3, −2); from C to (2, 5): (3, 2). Both signs have changed, so the angle is 180°.
- Write the full description. We have found every piece of information needed for this transformation.Rotation, 180°, centre (−1, 3)
Check. Rotate (−1, 1) about (−1, 3) by 180°: we get (−1, 5), which matches the vertex (−1, 5) of B. ✓
Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.
Method. Apply each transformation to the vertices of A and compare with B.
- Hannah’s reflection in the y-axis. Change the sign of every x-coordinate.(1, 1) → (−1, 1); (3, 1) → (−3, 1); (1, 4) → (−1, 4). This matches B.
- Ravi’s rotation of 180° about O. Change the sign of both coordinates.(1, 1) → (−1, −1), which is not a vertex of B.
- Decide. Only Hannah’s transformation produces B.Hannah is right.
Check. B is a mirror image of A (the right-angled corner is at the bottom right in B and the bottom left in A), so it is a reflection. ✓
Watch out. Test each description on every vertex, not just one that happens to match.