Grade 10 core mathematics Measurements and Geometry – Similarity and Enlargement Notes

Specific learning outcomes

1. (a) Identify and outline sub-sub-strands: Similarity, Enlargement, Scale factors, Application of Similarity and Enlargement.
2. (b) Determine the centre of enlargement and the linear scale factor for similar figures.
3. (c) Construct the image of an object under enlargement given the centre and the linear scale factor.
4. (d) Determine the area and volume scale factor of different figures and solids.
5. (e) Relate linear scale factor, area scale factor and volume scale factor in enlargements.
6. (f) Apply similarity and enlargement to real-life situations (maps, models, architecture, photography).
7. (g) Appreciate the use of similarity and enlargement in real-life situations.

Key concepts and definitions

• Similarity: Two figures are similar if corresponding angles are equal and corresponding sides are in the same ratio (linear scale factor).
• Enlargement (Dilation): A transformation from a centre O that increases (or decreases) distances from O by a constant factor k (the scale factor).
• Linear scale factor (k): Ratio of lengths in image to original: k = (image length) / (original length).
• Area scale factor: When lengths scale by k, areas scale by k².
• Volume scale factor: When lengths scale by k, volumes scale by k³.
• Centre of enlargement: The fixed point O from which every point of the original is connected by a ray and moved to a point on that ray.

Visual example (construction of an enlargement)

Example: Centre O(40,160), original A(80,120). Vector OA = (40,-40). With k = 1.5, A' = O + 1.5(OA) = (100,100). Same method for B,C gives B'(190,100), C'(145,10).

How to determine the centre of enlargement and linear scale factor

1. Given two similar figures on a plane with labelled corresponding vertices (e.g. A ↔ A', B ↔ B', C ↔ C'):
   1. Join A to A' and B to B'. The intersection of lines AA' and BB' is the centre of enlargement O (usually all AA', BB', CC' concur at O).
2. To find linear scale factor k:
   1. Measure OA' and OA (distances from O to corresponding vertices). Then k = OA' / OA.
   2. Alternatively k = length of corresponding image side / length of original side (e.g. A'B' / AB).
3. If k > 1 figure is an enlargement; if 0 < k < 1 figure is a reduction; k negative would produce an inversion through O plus scaling (advanced/optional topic).

Constructing an enlargement (compass/ruler method)

1. Given original figure and centre O and k:
   1. For each vertex P of the original, draw a ray OP.
   2. Measure distance OP (or use coordinates). Multiply OP by k to get OP'.
   3. On ray OP from O mark the point P' such that OP' = k · OP (use a ruler or compass).
   4. Join the new vertices P' to form the image.
2. Check: corresponding angles remain equal; corresponding sides are in ratio k.

Area and volume relationships

If linear scale factor is k:

• Area scale factor = k². (Example: if k = 3, area scales by 9.)
• Volume scale factor = k³. (Example: if k = 2, volume scales by 8.)
• Use these to compare surface areas and volumes of similar solids (important in model-making and maps).

Worked example: A cube of side 4 cm is enlarged by k = 1.5. New side = 6 cm, area scale = 1.5² = 2.25 so surface area ×2.25; volume scale = 1.5³ = 3.375 so volume ×3.375.

Real-life applications (Kenyan context)

• Maps: map scale (e.g. 1 cm : 5 km) uses linear scale factors to compute real distances on the ground. Convert map distances to field distances by multiplying by scale.
• Architecture & models: scale models of houses or classrooms use similarity to preserve proportions when building maquettes.
• Photography and printing: enlarging a photo changes area by k² (affects printing cost and paper use).
• Engineering & manufacturing: parts drawn at scale and then scaled up/down for production.
• Agriculture: scaled maps of farms to estimate area for planting; e.g. a field shown on a 1:10 000 map—use area factor to estimate real field area.

Suggested learning experiences / classroom activities

1. Hands-on enlargement:
   1. Give pupils small cut-out shapes (triangles/quadrilaterals), choose centre O on large paper, and use compass/ruler to create images for k = 0.5, 1.5 and 2. Compare results.
2. Find centre & k from pairs:
   1. Provide printed pairs of similar figures (unlabelled). Pupils join corresponding vertices to locate centre O and compute k by measuring distances.
3. Map exercise (Kenyan): use a local county map with known scale (e.g. 1:50 000). Measure the distance between towns on the map and compute real distances.
4. Model building: build a simple scale model of a classroom or a school gate using a chosen scale factor; compute areas and material needed using k² and k³.
5. Group presentation: research one real-life use of similarity (e.g. surveying, photography) and present how scale factors are important.

Materials: rulers, compasses, protractors, graph paper, scissors, printouts of shapes, local maps, calculators.

Suggested time: 2–3 lessons (one theory + one practical + assessment).

Practice questions

1. Given triangle ABC and its image A'B'C' shown on a sketch, lines AA' and BB' meet at O. Explain how to find k. (Answer: k = OA'/OA or k = A'B'/AB.)
2. Given centre O at (0,0). Point P is (2,3). If k = 3, find coordinates of P'. (Answer: P' = (6,9)).
3. A rectangular field 40 m by 25 m is represented on a plan with scale k = 1:200 (linear). What is the area of the field on the plan? What is the area scale factor? (Answer: plan lengths 0.2 m by 0.125 m; area scale factor = (1/200)² = 1/40 000.)
4. A solid model of a water tank has linear scale factor k = 0.5. If original tank volume is 8 m³, what is model volume? (Answer: model volume = 8 × 0.5³ = 8 × 0.125 = 1 m³.)
5. (Challenge) Two similar triangles have corresponding sides 6 cm and 15 cm. Find k and the ratio of their areas. (Answer: k = 15/6 = 2.5; areas ratio = k² = 6.25.)

Assessment ideas

• Short test: locate centres of enlargement for several pairs of figures, compute k, and construct images for given k values.
• Practical task: pupils make a scale model of a classroom element and report on linear, area and volume scale factors.
• Oral questioning: explain meaning of similarity and how k relates to areas and volumes.

Summary

Similarity and enlargement are powerful tools in geometry. Remember: linear scale factor k governs lengths, k² governs areas and k³ governs volumes. The centre of enlargement is found where lines joining corresponding points meet. Use these ideas when reading maps, making models or solving real-life measurement problems.