Similarity – Same Shape, Different Size
Similar shapes have the same shape but not necessarily the same size. They are like scaled copies of each other. Understanding similarity is key in map reading, scale models, photography, and many areas of engineering.
What Does Similar Mean?
Two shapes are similar if one can be obtained from the other by enlargement (scaling). All corresponding angles are equal and all corresponding sides are in the same ratio. The ratio is called the scale factor.
Congruence vs Similarity
| Property | Congruent | Similar |
|---|---|---|
| Same shape | Yes | Yes |
| Same size | Yes | Not necessarily |
| Angles equal | Yes | Yes |
| Sides equal | Yes | In proportion (same ratio) |
| Scale factor | Always 1 | Any positive value |
Scale Factor
The scale factor k = (length on image) / (length on original). If k > 1, the shape is enlarged. If k < 1, it is reduced. If k = 1, the shapes are congruent.
Each side of B is double A. Scale factor k = 6/3 = 8/4 = 10/5 = 2.
7 × 3 = 21 cm.
Similar Triangles – Conditions
| Condition | What It Means |
|---|---|
| AA (Angle-Angle) | Two pairs of equal angles (the third must then also match) |
| SSS (ratio) | All three pairs of sides are in the same ratio |
| SAS (ratio) | Two pairs of sides in ratio with the included angle equal |
Area and Volume Scale Factors
If the linear scale factor is k, then: Area scale factor = k². Volume scale factor = k³.
Area ratio = 3² = 9 : 1.
Volume ratio = 125/27. k³ = 125/27. k = ∛(125/27) = 5/3. Scale factor = 5 : 3.
Key Takeaways
- Similar shapes have the same angles and proportional sides.
- Scale factor k = image length / original length.
- Two triangles are similar if two angles match (AA condition).
- Area scales by k² and volume scales by k³.
Practice Questions
- Triangle X has sides 5, 12, 13. Triangle Y has sides 10, 24, 26. Are they similar? Find the scale factor.
- Two similar rectangles have widths 4 cm and 10 cm. The smaller has area 20 cm². Find the area of the larger.
- A model car is built to a scale of 1:25. The real car is 4.5 m long. Find the model length in cm.
- Two similar cylinders have radii in ratio 2:5. Find the ratio of their volumes.
- Triangles PQR and STU are similar. PQ = 9, QR = 12, ST = 6. Find TU.