GRADE 9 Mathematics GEOMETRY – SIMILARITIES AND ENLARGEMENTS Notes

Mathematics — Geometry: Similarities and Enlargements

Level: Kenyan lower secondary (age ≈ 14)

Learning objectives

• Understand what it means for two shapes to be similar.
• Use scale (enlargement) factor to construct or describe enlargements.
• Apply similarity tests (AA, SAS, SSS) to identify similar shapes.
• Use scale factor to find corresponding lengths and areas.

1. What is similarity?

Two plane shapes are similar if corresponding angles are equal and corresponding sides are in the same ratio (proportional). A similar shape is a scaled copy — it may be bigger or smaller, but has the same shape.

Key points

• Corresponding angles are equal (∠A = ∠A', ∠B = ∠B', ...).
• Corresponding sides are proportional: AB/A'B' = BC/B'C' = CA/C'A' = scale factor k.
• If k > 1 the image is an enlargement. If 0 < k < 1 it is a reduction.

2. Scale (enlargement) factor

Scale factor k = (length of image)/(length of original).

Example: If a side of a small triangle is 6 cm and the corresponding side on the large triangle is 15 cm, then k = 15/6 = 2.5.

3. Similarity tests

To check if two triangles are similar we use:

• AA (Angle-Angle): two equal angles ⇒ triangles similar.
• SAS (Side-Angle-Side): two sides in proportion and the included angle equal ⇒ triangles similar.
• SSS (Side-Side-Side): three corresponding sides in proportion ⇒ triangles similar.

4. Areas and scale factor

If shapes are similar with scale factor k, then area of image = k² × area of original.

Example: if k = 3 and original area = 8 cm², new area = 3² × 8 = 72 cm².

5. Simple diagrams (visual)

Below is an example of a triangle and its enlargement with scale factor k = 1.6 about the origin (0,0).

6. Working with coordinates

Enlargement about the origin (0,0): multiply each coordinate by k.

Example: Triangle with points A(1,1), B(3,1), C(1,4). If k = 2 (centre at origin), new points are:

• A' = (2×1, 2×1) = (2,2)
• B' = (2×3, 2×1) = (6,2)
• C' = (2×1, 2×4) = (2,8)

Enlargement about a point P (not origin): for each point X, do vector PX, multiply by k, then put the image at P + k·PX.

Short method: X' = P + k(X − P).

7. Worked examples

8. Quick tips and common mistakes

• Always match corresponding sides in the correct order before calculating a ratio.
• Scale factor is image ÷ original (not the other way round).
• Angles must match for similarity — if two pairs of angles match, the third must also match.
• Area scale = k². Do not use k for area directly.

9. Practice questions (with answers)

1. Two similar triangles have side lengths 8 cm and 20 cm for a corresponding pair. Find k and the missing side if the other side of the small triangle is 6 cm.
   Answer: k = 20/8 = 2.5. The corresponding large side = 6 × 2.5 = 15 cm.
2. A rectangle of size 4 cm by 7 cm is enlarged by k = 0.5. What are the new dimensions and area?
   Answer: New dimensions: 2 cm by 3.5 cm. Original area = 28 cm², new area = k² × 28 = 0.25 × 28 = 7 cm².
3. Triangle ΔABC has A(1,2), B(4,2), C(1,6). Find coordinates after enlargement with centre at origin and k = 3.
   Answer: A'=(3,6), B'=(12,6), C'=(3,18).
4. Given triangles are similar. If two corresponding sides are 9 cm and 15 cm, and the area of the smaller is 54 cm², what is the area of the larger?
   Answer: k = 15/9 = 5/3. Area scale = (5/3)² = 25/9. New area = 54 × 25/9 = 6 × 25 = 150 cm².

10. Classroom/Exam advice

• Label corresponding vertices (A with A', B with B') so you do not mix sides.
• Draw clear diagrams and write the scale factor k on them.
• For coordinate enlargements, practise both centres at origin and other points.

These notes give the essentials on similarity and enlargements. Practise with real sheet work: draw triangles, measure sides and angles, and check proportions — this builds confidence for KCSE style problems.