SECONDARY 2 MATHEMATICS CLASSROOM · CHAPTER 7 · CONGRUENCE · SIMILARITY · ENLARGEMENT · G2/G3
Congruence, Similarity and Enlargement: When Shape Stays but Size Changes
Congruent figures have the same shape and the same size. Similar figures have the same shape but may have different sizes. Enlargement turns correspondence into a controlled multiplication.
Chapter 6 studied quadratic structure: the same expression could be expanded, factorised, solved or graphed depending on the mathematical job. Chapter 7 changes the objects but keeps the discipline. In geometry, the first task is not calculation. It is to identify which points, sides and angles correspond, decide whether the figures are congruent or similar, and only then apply the correct scale relationship.
Classroom rule: establish correspondence → identify the relationship → state the scale factor in the correct direction → use matching quantities → square the factor for area → cube it for volume → verify the answer against the diagram and context.
Level boundary. The shared G2/G3 foundation in this classroom is correspondence, congruence, similarity, enlargement and proportional reasoning. Where a school’s current sequence treats area/volume scale-factor transfer or more complex multi-stage similarity as later-course work, those sections should be used as bridge material rather than forced into the present lesson. The mathematics remains connected; the timing may differ.
Official reference: MOE G2 and G3 Mathematics Syllabuses.
Navigate: retrieval · correspondence · congruence · similarity · scale factors · area and volume · enlargement · centre of enlargement · models · misconceptions · guided practice · assessment transfer · exit ticket.
Featured Answer: What Is the Difference Between Congruence and Similarity?
Congruent figures have equal corresponding lengths and equal corresponding angles. Similar figures have equal corresponding angles and proportional corresponding lengths. Every pair of congruent figures is also similar with scale factor 1, but similar figures do not have to be congruent.
Congruent: same shape, same size. Similar: same shape, proportional size. Enlargement: a transformation that multiplies distances from a centre by one scale factor.
How to Use This Classroom
- Mark corresponding vertices before writing ratios.
- Keep the order of corresponding points consistent.
- Decide whether the question is about congruence, similarity or transformation.
- Write the scale factor with direction: image ÷ object or new ÷ original.
- Use length scale factors only with lengths.
- Use the square of the length factor with areas.
- Use the cube of the length factor with volumes.
- For an enlargement, trace rays through corresponding points to locate the centre.
- Check whether the numerical answer is sensible for an enlargement or reduction.
- Return to the diagram and state the requested quantity with units.
1. Retrieval: Ratio Is the Language Under Similarity
If two matching lengths are 6 cm and 9 cm, the scale factor from the first figure to the second is 9/6=3/2. That single multiplicative relationship is the engine behind the whole chapter.
2. Multiplication, Not Addition
Similar figures do not grow by adding the same amount to every side. A 4 cm side becoming 6 cm and an 8 cm side becoming 10 cm does not preserve shape because the multipliers are different: 6/4≠10/8.
3. Quick Retrieval Diagnostic
- Simplify 12:18.
- Find the multiplier from 8 to 14.
- If a length is multiplied by 3, what happens to its square?
- If a model is at scale 1:50, what does 1 cm on the model represent?
Answers
2:3. 14/8=7/4. Its square is multiplied by 9. 50 cm in the represented object.
4. Correspondence Comes Before Calculation
Suppose triangle ABC corresponds to triangle PQR. If A↔P, B↔Q and C↔R, then AB↔PQ, BC↔QR and AC↔PR. Ratios must compare matching sides in that same order.
5. Vertex Order Is Information
Writing △ABC∼△PQR is not decorative notation. It tells us exactly which vertices correspond. If the order changes, the implied side pairings change too.
6. Teacher Model 1: Build the Correspondence Table
| First figure | Second figure |
|---|---|
| A | P |
| B | Q |
| C | R |
| AB | PQ |
| BC | QR |
| AC | PR |
7. Rotated Figures Still Correspond
A figure can be turned, reflected or drawn on another part of the page. Orientation does not decide similarity or congruence. Corresponding geometry does.
8. Congruent Figures Preserve Every Length
If two figures are congruent, all corresponding side lengths and angles are equal. The scale factor is 1.
9. Congruence Is Stronger Than Equal Area
Two rectangles can both have area 24 cm² while one is 4 by 6 and another is 3 by 8. Equal area alone does not prove congruence.
10. Congruence Is Stronger Than Equal Perimeter
Different shapes can have the same perimeter. Congruence requires the entire corresponding structure to match.
11. Teacher Model 2: Congruent Triangles
If △ABC≅△PQR and AB=7 cm, BC=9 cm, AC=11 cm, then PQ=7 cm, QR=9 cm and PR=11 cm. No scale calculation is needed because the scale factor is 1.
12. Congruent Angles Match Too
If ∠B=63°, then the corresponding angle ∠Q=63°.
13. Similar Figures Preserve Shape Through Proportion
For similar figures, corresponding angles are equal and corresponding side lengths are in one constant ratio.
14. Teacher Model 3: Detect Similarity From Side Ratios
One triangle has sides 4, 6 and 8. Another has sides 6, 9 and 12.
6/4=9/6=12/8=3/2.
All corresponding sides share the same multiplier, so the side data are consistent with similarity.
15. One Matching Ratio Is Not Enough
If only one pair of sides has ratio 2:3, that does not prove the entire figures are similar. The whole relevant correspondence must satisfy the required geometric conditions.
16. Equal Corresponding Angles Control Shape
Similarity is not merely about scaled lengths. The angle structure must also be preserved.
17. Scale Factor Has Direction
If a 5 cm side in the first figure corresponds to an 8 cm side in the second, the scale factor from first to second is 8/5. The scale factor from second back to first is 5/8.
18. Teacher Model 4: Find a Missing Length
Two similar triangles have corresponding sides 6 cm and 15 cm. Another side on the smaller triangle is 8 cm. Find the corresponding larger side.
Scale factor small→large =15/6=5/2.
Larger side=8×5/2=20 cm.
19. Teacher Model 5: Work Backward
A larger figure has side 27 cm corresponding to 18 cm on a smaller figure. Another larger side is 21 cm.
Large→small factor=18/27=2/3.
Smaller side=21×2/3=14 cm.
20. Ratio Tables Can Stabilise Multi-Step Questions
| Small | Large |
|---|---|
| 6 | 15 |
| 8 | 20 |
21. Algebra Can Represent an Unknown Corresponding Length
If 7/10=x/25, then x=17.5. The geometry supplies the ratio; algebra completes the calculation.
22. Teacher Model 6: Similarity With an Algebraic Side
Suppose corresponding sides are 4 and 10, while another pair is x+1 and 15.
10/4=15/(x+1).
2.5(x+1)=15, so x+1=6 and x=5.
Your Turn 1 — Correspondence and Length Scale
- A 7 cm side corresponds to 10.5 cm. Find the scale factor.
- A smaller side is 12 cm and the factor small→large is 5/4. Find the larger side.
- A larger side is 35 cm and the factor small→large is 7/5. Find the smaller side.
- Explain why adding 3 cm to every side does not generally produce a similar figure.
Answers
3/2. 15 cm. 25 cm. Similarity requires one constant multiplicative factor, not a constant additive increase.
23. Length Scale Factor Does Not Apply Directly to Area
If every length is multiplied by k, two independent dimensions of an area are each multiplied by k. Therefore area is multiplied by k².
24. Teacher Model 7: Area Scale Factor
If the length scale factor is 3, the area scale factor is 3²=9.
25. Reverse Area Problems Need a Square Root
If two similar figures have area ratio 49:81, their corresponding length ratio is √49:√81=7:9.
26. Teacher Model 8: Find a Similar Area
A smaller similar figure has area 24 cm². The length factor to the larger figure is 3/2.
Area factor=(3/2)²=9/4.
Larger area=24×9/4=54 cm².
27. Volume Uses Three Dimensions
For similar solids, if the length factor is k, the volume factor is k³.
28. Teacher Model 9: Volume Scale Factor
If the length factor is 2, the volume factor is 2³=8.
29. Reverse Volume Problems Need a Cube Root
If the volume ratio is 64:125, the corresponding length ratio is ∛64:∛125=4:5.
30. Area and Volume Transfer Are Bridge Material Where Required
If this relationship has not yet been introduced in a student’s current school sequence, treat Sections 23–29 as structured preparation. The essential shared idea remains the same: dimensions determine the power applied to the scale factor.
Your Turn 2 — Area and Volume Scale
- Length factor 4. Find the area factor.
- Length factor 3/5. Find the area factor.
- Area ratio 25:64. Find the corresponding length ratio.
- Length factor 3. Find the volume factor.
- Volume ratio 8:27. Find the corresponding length ratio.
Answers
16. 9/25. 5:8. 27. 2:3.
31. Enlargement Is a Transformation With a Centre and Scale Factor
Under an enlargement with centre O and scale factor k, each image point lies on the same ray from O as its original point, and its distance from O is multiplied by k.
32. Scale Factor Greater Than 1 Produces a Larger Image
If k=2, every image point is twice as far from the centre as its corresponding original point.
33. Scale Factor Between 0 and 1 Produces a Reduction
If k=1/2, every image point lies halfway from the centre to the corresponding original point.
34. Scale Factor 1 Leaves Size Unchanged
Every point remains at the same distance from the centre. The image coincides with the object.
35. Teacher Model 10: Coordinate Enlargement From the Origin
With centre (0,0) and scale factor 3, point A(2,−1) maps to A′(6,−3). Both coordinates are multiplied by 3 because the origin is the centre.
36. Do Not Multiply Coordinates Blindly When the Centre Is Not the Origin
The enlargement acts on displacement from the centre, not on raw coordinates.
37. Teacher Model 11: Non-Origin Centre
Centre C=(1,2), point A=(4,6), scale factor 2.
Vector from C to A is (3,4). Double it to (6,8). Add back to the centre: A′=(7,10).
Coordinate enlargement routine: subtract centre → multiply displacement by k → add centre back.
38. The Centre of Enlargement Lies on Lines Through Corresponding Points
Join A to A′ and B to B′. Extend those lines if necessary. Their intersection is the centre of enlargement.
39. One Corresponding Pair Gives a Line, Not a Unique Centre
A single object-image point pair only tells us that the centre lies somewhere on the line through them. A second pair is needed to identify the intersection uniquely.
40. Teacher Model 12: Determine the Scale Factor From Distances to the Centre
If OA=4 cm and OA′=10 cm along the same ray, the scale factor is 10/4=5/2.
41. The Image Must Preserve Corresponding Angles
An enlargement changes size but preserves shape, so the object and image are similar.
42. Maps and Scale Drawings Are Similarity in Context
A scale of 1:500 means one unit on the drawing represents 500 of the same units in reality. Units must be aligned before multiplication.
43. Teacher Model 13: Floor Plan
A floor plan uses scale 1:100. A wall measures 8.4 cm on the plan.
Actual length=8.4×100=840 cm=8.4 m.
44. Teacher Model 14: Model Vehicle
A model car is built at scale 1:24. The real car is 4.32 m long.
4.32 m=432 cm. Model length=432/24=18 cm.
45. Unit Conversion Must Not Be Hidden Inside the Scale Ratio
The ratio compares like units. Convert first or explicitly track the units through the calculation.
46. Similarity Can Model Indirect Measurement
When two triangles are known to be similar, an inaccessible height or distance can be found from proportional corresponding lengths. The important step is proving or being given the similarity before writing the proportion.
47. Misconception Clinic: Same Shape Means Congruent
Repair: same shape with different size is similarity, not congruence.
48. Misconception Clinic: Same Area Means Congruent
Repair: many non-congruent figures can share the same area.
49. Misconception Clinic: Use Whatever Side Looks Closest
Repair: proximity on the page is irrelevant. Use corresponding vertices and sides.
50. Misconception Clinic: Reverse One Ratio Halfway Through
Repair: keep one direction throughout the equation.
51. Misconception Clinic: Length Factor 3 Means Area Factor 3
Repair: area has two dimensions, so factor 3 becomes area factor 9.
52. Misconception Clinic: Length Factor 3 Means Volume Factor 9
Repair: volume has three dimensions, so the factor is 27.
53. Misconception Clinic: Enlargement Always Means Bigger
In transformation language, an enlargement can have a scale factor between 0 and 1 and therefore produce a smaller image.
54. Misconception Clinic: Multiply Coordinates by k for Every Centre
Repair: this shortcut only works directly when the centre is the origin. Otherwise multiply the displacement from the stated centre.
55. Misconception Clinic: Diagram Appearance Proves Similarity
Repair: diagrams may not be drawn to scale. Use stated lengths, angles and geometric conditions.
56. Guided Practice A: Congruence or Similarity?
- Two squares have side lengths 5 cm and 5 cm.
- Two squares have side lengths 5 cm and 8 cm.
- Two rectangles are 4×6 and 6×9.
- Two rectangles are 4×6 and 5×7.
Solutions
Congruent. Similar but not congruent. Similar because both dimensions scale by 3/2. Not similar because 5/4≠7/6.
57. Guided Practice B: Missing Lengths
- 4 cm corresponds to 10 cm. What does 7 cm correspond to?
- 15 cm corresponds to 9 cm. What does 25 cm correspond to in the second figure?
- A length factor is 1.6. Find the image of 12.5 cm.
Solutions
17.5 cm. 15 cm. 20 cm.
58. Guided Practice C: Area and Volume
- Length factor 5/2. Find area factor.
- Area ratio 36:121. Find length ratio.
- Length factor 4/3. Find volume factor.
- Volume ratio 125:216. Find length ratio.
Solutions
25/4. 6:11. 64/27. 5:6.
59. Guided Practice D: Enlargement Coordinates
- Centre origin, k=2, P(−3,4).
- Centre origin, k=1/2, Q(8,−6).
- Centre (1,1), k=3, R(3,4).
Solutions
P′(−6,8). Q′(4,−3). From centre to R is (2,3); triple is (6,9); R′=(7,10).
60. Guided Practice E: Scale Drawing
A map has scale 1:25 000. Two points are 6.8 cm apart on the map. Find the real distance in kilometres.
Worked solution
6.8×25 000=170 000 cm=1 700 m=1.7 km.
61. Challenge Practice: Two-Stage Similarity
Figure A scales to B by factor 3/2. Figure B scales to C by factor 4/3. Find the direct factor from A to C.
Answer
(3/2)(4/3)=2. Scale factors compose multiplicatively.
62. Challenge Practice: Reverse the Dimension
Two similar solids have volume ratio 343:1000. Find their corresponding length ratio.
Answer
∛343:∛1000=7:10.
63. Assessment Method: Name the Relationship Before Writing an Equation
Congruent, similar and enlarged are not interchangeable labels. The relationship determines the equation.
64. Assessment Method: Annotate Correspondence
Mark matching vertices and sides on the diagram before inserting numbers into a ratio.
65. Assessment Method: State the Direction of k
Writing “small→large k=5/3” prevents many reciprocal errors.
66. Assessment Method: Match Dimension to Power
- length → k;
- area → k²;
- volume → k³.
67. Assessment Method: Check Magnitude
If k>1, a corresponding image length should be larger. If 0<k<1, it should be smaller. This quick check catches many arithmetic reversals.
68. Assessment Method: Units Belong to the Final Interpretation
Lengths use linear units, areas use square units and volumes use cubic units.
69. Oral Classroom Check
- What is the difference between congruence and similarity?
- Why is correspondence established before scale factor?
- What does vertex order tell you?
- Why does scale factor have direction?
- If the length factor is 4, what is the area factor?
- If the length factor is 4, what is the volume factor?
- How do you locate a centre of enlargement?
- Why can we multiply coordinates directly only when the centre is the origin?
- Why does equal area not prove congruence?
- Why might some scale-factor transfer sections be used as bridge work for a particular G2 sequence?
70. Exit Ticket
- State the difference between congruent and similar figures.
- A side of 8 cm corresponds to 14 cm. Find the scale factor.
- A 12 cm side is enlarged by factor 7/4. Find its image length.
- Length factor 3. Find area factor.
- Length factor 3. Find volume factor.
- Area ratio 16:49. Find length ratio.
- Centre origin, k=2, map (−4,3).
- Centre (2,1), k=2, map (5,5).
- Explain why a diagram that “looks similar” is insufficient evidence.
- At scale 1:200, what real length does 3.5 cm represent?
Exit-ticket solutions
Congruent means same shape and size; similar means same shape with proportional lengths. 14/8=7/4. 21 cm. 9. 27. 4:7. (−8,6). Centre-to-point displacement is (3,4); doubled is (6,8), so image is (8,9). Appearance is unreliable and diagrams may not be drawn to scale; correspondence and geometric conditions are required. 700 cm=7 m.
71. Homework: Retrieval, Variation and Transfer
Layer 1 — Retrieval
- classify four pairs of figures as congruent, similar or neither;
- write three correspondence tables;
- find five missing lengths from similarity;
- reverse two scale factors;
- complete one map-scale conversion.
Layer 2 — Variation
- solve two algebraic similarity questions;
- solve two area-factor questions;
- solve two volume-factor questions where appropriate to level;
- perform three coordinate enlargements from different centres;
- locate one centre of enlargement from two corresponding point pairs.
Layer 3 — Transfer
Choose a real object that has a scale representation—a map, floor plan, model vehicle, architectural drawing or digital image. State the scale relationship and explain which quantities scale linearly, quadratically or cubically.
72. The Seven-Day Return Cycle
- Day 0: correspondence, congruence and similarity.
- Day 1: missing lengths and one reversed scale factor.
- Day 3: mixed length/area/volume scale without headings.
- Day 7: changed exit ticket with one transformation and one real-world scale problem.
73. A 60-Minute Teaching Lesson
- 10 minutes: ratio retrieval and correspondence.
- 10 minutes: congruence versus similarity.
- 15 minutes: missing lengths and scale-factor direction.
- 10 minutes: area/volume transfer where appropriate.
- 10 minutes: enlargement and centre.
- 5 minutes: exit ticket.
74. A 90-Minute Teaching Lesson
- 15 minutes: retrieval and correspondence diagnostics.
- 15 minutes: congruence and similarity conditions.
- 20 minutes: length scale and algebraic missing lengths.
- 15 minutes: area and volume scale-factor transfer.
- 15 minutes: enlargement, coordinates and centre.
- 5 minutes: real-world scale drawing.
- 5 minutes: exit ticket and return date.
75. The Full Similarity Routine
identify corresponding points → confirm similarity → choose one matching pair → calculate directed scale factor → apply to matching quantity → verify proportion → state units.
76. The Full Enlargement Routine
identify centre → identify scale factor → find displacement from centre → multiply displacement → place image point → repeat for all vertices → check shape and correspondence.
77. Connect Back to Chapter 6
Return to Secondary 2 Chapter 6: Quadratic Expressions, Equations, Functions and Graphs when algebraic manipulation, ratio equations or coordinate substitution is unstable. Chapter 7 changes the visual object but keeps the same habits: identify structure, preserve equivalence, apply the correct operation and verify the result.
78. Specialist Companions
- Secondary 2 Mathematics Learning Guide | Geometry, Similarity and Mathematical Constraints
- Secondary 2 Mathematics Learning Guide | Coordinate Geometry, Distance, Midpoint and Locus
- Secondary 2 Mathematics Learning Guide | Right-Angled Triangle Trigonometry, Angles and Missing Lengths
79. Why This Chapter Matters for Chapter 8
Similarity teaches that corresponding side ratios carry geometric information. Chapter 8 uses that idea inside right-angled triangles. Instead of comparing two drawn similar figures explicitly, trigonometric ratios package repeated right-triangle similarity into sine, cosine and tangent. The discipline remains the same: identify the triangle, identify the relevant sides, choose the correct ratio and check whether the answer fits the geometry.
80. Ready for Chapter 8?
You are ready to continue when the material appropriate to your current level is stable.
- identify corresponding vertices, sides and angles;
- distinguish congruence from similarity;
- calculate and reverse a directed length scale factor;
- find missing corresponding lengths;
- connect length factor to area and volume factor where appropriate to the current course sequence;
- perform an enlargement from a stated centre and scale factor;
- locate a centre of enlargement from corresponding point pairs;
- solve basic scale-drawing and map problems with correct unit conversion.
If one item is weak, return to the smallest section that owns it and solve a changed example. When the appropriate set is stable, continue to Chapter 8: Right-Angled Triangle Trigonometry, Angles and Missing Lengths.
