Secondary Mathematics · Worked Repair Guide 23
A transformation changes a figure according to a rule. Some transformations preserve length and angle; some preserve shape but change size; some reverse orientation; some move every point by the same vector. The strongest way to understand them is not to memorise four unrelated procedures but to ask what information is preserved and what changes.
This guide develops one central habit: name the transformation completely, then check its invariants. A reflection needs a mirror line. A rotation needs centre, angle and direction. An enlargement needs centre and scale factor. A translation needs a displacement vector. Missing parameters make an answer incomplete even when the picture looks right.
All coordinate examples are original teaching material. The conceptual owner remains How Secondary 1 Transformations and Symmetry Work; this page adds a deeper worked-repair route across coordinates and proof.
1. A transformation is a rule from points to points
If point P maps to P′, every point of the original figure has an image under the same transformation rule. The shape’s image is built point by point.
The prime symbol does not mean “larger” or “newer”. It labels the image. If A maps to A′, B to B′ and C to C′, the triangle ABC maps to triangle A′B′C′.
Entry check: if translation vector is (4,−3), then (2,5) maps to (6,2). Applying the inverse vector (−4,3) returns the image to (2,5).
2. Translation adds the same vector to every point
A translation by vector (a,b) sends (x,y) to (x+a,y+b).
For vector (−3,5), A=(4,1) maps to A′=(1,6); B=(8,−2) maps to B′=(5,3).
The displacement A→A′ and B→B′ is identical. Therefore corresponding segments remain parallel and equal in length.
A translation preserves distance, angle, area and orientation. It changes position but not size or handedness.
3. Recover a translation vector from one point pair
If P=(−2,4) maps to P′=(5,1), the vector is image minus original: (5−(−2),1−4)=(7,−3).
Check a second vertex if available. If Q=(3,6) maps under the same translation, it should become (10,3).
If different vertices require different displacement vectors, the transformation is not that single translation.
This is a direct application of vector difference from the Batch 05 vector guide.
4. Reflection needs a mirror line
A reflection maps each point to the opposite side of a mirror line at equal perpendicular distance. The mirror line is the perpendicular bisector of the segment joining each point to its image.
Across the y-axis, (x,y) maps to (−x,y). Across the x-axis, (x,y) maps to (x,−y).
Across y=x, coordinates swap: (x,y)→(y,x). Across y=−x, (x,y)→(−y,−x).
These coordinate rules are consequences of the mirror lines, not arbitrary sign tricks.
5. Reflecting in vertical or horizontal lines needs distance from that line
Reflect P=(7,2) in x=3. The point is 4 units to the right of x=3, so its image lies 4 units to the left: x=−1. Thus P′=(−1,2).
Reflect Q=(−2,5) in y=1. It is 4 units above the line, so the image is 4 units below: Q′=(−2,−3).
Do not merely negate a coordinate unless the mirror line is the corresponding axis. Reflecting in x=3 is not the same as reflecting in the y-axis.
6. Reflection preserves lengths but reverses orientation
A triangle labelled A-B-C clockwise will appear A′-B′-C′ anticlockwise after reflection, assuming the labels follow corresponding vertices.
Lengths and angles remain equal, so the image is congruent to the original. Area is preserved.
Orientation reversal is a useful diagnostic. If a supposed reflection leaves the ordering clockwise and no other relabelling is involved, recheck the construction.
A translation or rotation preserves orientation; a reflection reverses it.
7. Rotation needs centre, angle and direction
A rotation turns every point by the same angle about one fixed centre. Distances from the centre are preserved.
About the origin, a 90° anticlockwise rotation maps (x,y)→(−y,x). A 90° clockwise rotation maps (x,y)→(y,−x). A 180° rotation maps (x,y)→(−x,−y).
For P=(3,−2), 90° anticlockwise gives (2,3). Check radius from origin: both have squared distance 13.
Writing only “rotation 90°” is incomplete because clockwise and anticlockwise differ unless the angle is 180°.
8. Rotate about a centre other than the origin by translating the centre to zero
Rotate P=(5,4) by 90° anticlockwise about C=(2,1).
First subtract the centre: relative vector CP=(3,3). Rotate this vector to (−3,3). Add the centre back: P′=(−1,4).
Check distances: CP=3√2 and CP′ also 3√2.
This “subtract centre → transform → add centre” method generalises many coordinate transformations about non-origin centres.
9. Enlargement needs centre and scale factor
An enlargement with centre C and scale factor k maps vector CP to CP′=k·CP.
With centre origin and scale factor 3, (2,−1) maps to (6,−3). Lengths triple; areas multiply by 9.
With scale factor 1/2, the image lies halfway from centre to original point. With k=1, every point stays fixed.
Unlike translation, rotation and reflection, enlargement does not generally preserve length or area.
10. Negative scale factors cross the centre
Where negative scale factors are in scope, k=−2 sends each point to the opposite side of the centre and doubles its distance from the centre.
About the origin, P=(3,1) maps to P′=(−6,−2).
The negative sign changes direction from the centre; the magnitude |k| changes size.
A negative enlargement is not simply a reflection in an arbitrary line. It is a central scaling and can be viewed as a 180° turn combined with a positive enlargement about the same centre.
11. Enlargement preserves shape and angles but scales lengths
For non-zero k, an enlargement produces a similar image. Corresponding angles are equal and corresponding lengths are in ratio |k|.
Area scales by k². If |k|=3, area multiplies by 9. For three-dimensional similar objects, volume would scale by |k|³.
This connects directly to Angles, Similarity and Geometric Reasoning and Units, Scale and Measurement.
12. Fixed points reveal the transformation
A point on a reflection line remains fixed under that reflection. The centre of a rotation remains fixed. The centre of an enlargement remains fixed for any scale factor.
A non-zero translation has no fixed point because every point moves by the same non-zero vector.
These facts help distinguish transformations from partial information. If a claimed non-zero translation leaves one vertex fixed, something is wrong.
Fixed sets can also be larger: every point on the mirror line is fixed, whereas a non-trivial rotation usually fixes only its centre in the plane.
13. Lines of symmetry are reflection invariants of the whole figure
A line of symmetry is a mirror line that maps the entire figure onto itself.
A non-square rectangle has two lines of symmetry through midpoints of opposite sides. A square has four. A general parallelogram has none.
Do not count lines that merely pass through the centre. The reflection must reproduce the complete figure.
For irregular figures, test corresponding features on both sides rather than relying on apparent balance.
14. Rotational symmetry is measured by order
A figure has rotational symmetry of order n if it maps onto itself n times during one full 360° turn, including the starting position.
A non-square rectangle has order 2; a square order 4; an equilateral triangle order 3.
The smallest positive rotation angle is 360°/n.
A generic shape has order 1 because only a full 360° turn maps it onto itself.
15. Composition means order matters
Reflecting a point in the x-axis and then translating by (2,0) generally differs from translating first and reflecting second.
Take P=(1,3). Reflect first: (1,−3), then translate: (3,−3).
Translate first: (3,3), then reflect: (3,−3). In this particular case the results happen to agree because the translation is parallel to the mirror line.
Now use translation (0,2). Reflect then translate gives (1,−1); translate then reflect gives (1,−5). The order matters.
16. Two reflections can create a translation or rotation
As an optional extension, reflecting in two parallel lines produces a translation perpendicular to those lines, with displacement twice their separation.
Reflecting in two intersecting lines produces a rotation about their intersection by twice the directed angle between the lines.
This explains why combinations of simple transformations form a larger structure rather than a random collection of moves.
Do not use these composition results without checking the order of the mirror lines.
17. Transformation matrices encode origin-centred linear rules
As enrichment, a 90° anticlockwise rotation about the origin can be represented by the matrix [[0,−1],[1,0]], which maps column vector (x,y) to (−y,x).
Reflection in the x-axis uses [[1,0],[0,−1]]. Enlargement by factor k about the origin uses [[k,0],[0,k]].
A non-zero translation cannot be represented by an ordinary 2×2 matrix acting on position vectors alone because such matrices always send the origin to the origin.
This fixed-origin observation distinguishes linear transformations from affine translations.
18. Invariants are powerful checking tools
Translation: preserve lengths, angles, area, orientation, parallelism.
Rotation: preserve lengths, angles, area, orientation.
Reflection: preserve lengths, angles and area, but reverse orientation.
Enlargement: preserve angles and shape; scale lengths and area.
If a translated triangle has a different side length from its original, the coordinate calculation must be wrong. Invariants can disagree with the working and therefore act as independent checks.
19. Capstone: identify a transformation from coordinates
Triangle A(1,1), B(4,1), C(1,3) maps to A′(−1,1), B′(−1,4), C′(−3,1).
Test A: (1,1)→(−1,1). A 90° anticlockwise rotation about the origin would give (−1,1).
Test B: (4,1)→(−1,4), also matching (x,y)→(−y,x). Test C: (1,3)→(−3,1), again matching.
Therefore the complete transformation is a rotation 90° anticlockwise about the origin.
Side lengths and orientation provide independent checks: lengths are preserved and the orientation remains unchanged.
20. Independent practice
- Translate (3,−2) by vector (5,4).
- Find the translation vector taking (−1,6) to (4,−2).
- Reflect (5,−3) in the y-axis.
- Reflect (5,−3) in the x-axis.
- Reflect (2,7) in y=x.
- Reflect (7,4) in x=2.
- Rotate (4,1) 90° anticlockwise about the origin.
- Rotate (4,1) 180° about the origin.
- Rotate (5,4) 90° anticlockwise about centre (2,1).
- Enlarge (3,−2) by factor 4 about the origin.
- Enlarge (5,3) by factor 1/2 about centre (1,1).
- Where in scope, enlarge (2,−1) by factor −3 about the origin.
- State whether reflection preserves orientation.
- State the rotational symmetry order of a regular hexagon.
- State the smallest positive rotation mapping a square to itself.
- A transformation preserves all lengths and reverses orientation. Name a likely transformation type.
- A non-zero transformation leaves every point on one straight line fixed and swaps the two sides. Name it.
- Find the image of (2,3) after reflection in the x-axis then translation by (0,5).
- Reverse the order in Question 18 and compare.
- Explain why an enlargement of scale factor 3 multiplies area by 9.
21. Worked answers
1. (8,2).
2. (5,−8). Image minus original.
3. (−5,−3).
4. (5,3).
5. (7,2). Swap coordinates.
6. (−3,4). The original x=7 is 5 units right of x=2; the image is 5 units left at x=−3.
7. (−1,4).
8. (−4,−1).
9. (−1,4). Relative vector from centre is (3,3); rotate to (−3,3), then add centre.
10. (12,−8).
11. (3,2). Relative vector (4,2) scales to (2,1), then add centre.
12. (−6,3).
13. No. Reflection reverses orientation.
14. 6.
15. 90°.
16. Reflection. A reflection is an isometry that reverses orientation.
17. Reflection in that line.
18. (2,2). Reflection gives (2,−3), then add (0,5).
19. (2,−8). Translate first to (2,8), then reflect in x-axis. The results differ, so order matters.
20. Area contains two independent length dimensions, each multiplied by 3, so the combined factor is 3²=9.
22. Diagnose transformation errors by missing parameters and invariants
Common failures include naming “rotation” without centre or direction, reflecting in x=3 as though it were the y-axis, applying an enlargement scale factor directly to global coordinates when the centre is elsewhere, or performing two transformations in the wrong order.
A useful correction note names the missing structure: “translate by one common vector”, “mirror line is the perpendicular bisector”, “subtract centre before origin-based rule”, or “length must be invariant under an isometry”.
Then test a second vertex. One matching point rarely proves the complete transformation; all points must follow the same rule.
23. Continue through the BTT learning routes
Return to the BTT Mathematics Hub or BTT Mathematical Lab. Use Angles, Similarity and Geometric Reasoning for geometric conditions and Vectors, Magnitude, Direction and Geometric Reasoning for translations and directed movement.
Within Batch 06, use Coordinate Geometry to analyse images numerically, Sequences, Patterns and Nth-Term Generalisation for repeated transformation patterns, and Direct and Inverse Proportion, Variation and Rates for enlargement scale relationships.
24. Sources and scope
The coordinate images, transformation compositions, symmetry examples and practice questions are original teaching material. Invariants are used as mathematical checks on the transformation definitions.
For the current Singapore Secondary curriculum doorway, see MOE: Curriculum for secondary schools. Match negative scale factors, matrices and composition extensions to the learner’s actual subject level and school programme.