Same shape, different size
Similar figures have the same shape but not necessarily the same size. The symbol is ∼.
Two figures are similar when:
- Corresponding angles are equal, and
- Corresponding sides are proportional (all in the same ratio).
That ratio is the scale factor.
Example
Triangle ABC has sides 3, 4, 5. Triangle DEF has sides 6, 8, 10.
6/3 = 8/4 = 10/5 = 2 → similar, scale factor 2.
Proving triangles similar
| Test | Need |
|---|---|
| AA similarity | two pairs of equal angles (the third follows automatically) |
| SSS similarity | all three side ratios equal |
| SAS similarity | two side ratios equal and the angle between them equal |
AA is the most used.
Finding missing sides
Set up a proportion with corresponding sides.
Example: △PQR ∼ △XYZ. PQ = 6, QR = 9, XY = 10. Find YZ.
- 6/10 = 9/YZ
- 6 YZ = 90 → YZ = 15
Similar vs congruent
| Congruent (≅) | Similar (∼) |
|---|---|
| same shape and size | same shape |
| scale factor 1 | any scale factor |
Areas and volumes
If lengths have scale factor k:
- Areas scale by k²
- Volumes scale by k³
A photo enlarged by 3 has 9 times the area.
Indirect measurement
Similar triangles let you measure things you can’t reach. This is indirect measurement.
Shadow method: A 1.6 m student casts a 2 m shadow. A flagpole casts a 15 m shadow at the same time.
1.6/2 = h/15 → h = 12 m
Mirror method: Place a mirror on the ground and step back until you see the top of a tree. The angles of incidence and reflection are equal, so the triangles are similar.
Nested triangles
When a line parallel to one side cuts a triangle, it creates a smaller similar triangle (by AA, using corresponding angles).
Example: In △ABC, DE ∥ BC with D on AB and E on AC. AD = 4, AB = 10, BC = 15. Then DE/BC = AD/AB → DE = 15 × 4/10 = 6.
Worked examples
Example 1: Are rectangles 4 × 6 and 6 × 9 similar? 6/4 = 1.5 and 9/6 = 1.5 → yes.
Example 2: Two triangles have angles 40°, 60°, 80° and 40°, 80°, 60°. Similar? Yes (AA).
Example 3: Scale factor 2.5: a side of 8 cm becomes? 20 cm.
Common mistakes
- Matching the wrong sides. Match sides opposite equal angles.
- Assuming all rectangles are similar. Check the side ratios.
- Scaling area by k instead of k².
Try it yourself
- Are triangles 5, 12, 13 and 10, 24, 26 similar?
- △ABC ∼ △DEF; AB = 4, DE = 10, BC = 7. Find EF.
- A 2 m post casts a 3 m shadow; a building casts a 36 m shadow. Height?
- Two similar figures have scale factor 4. Ratio of areas?
Answers: 1) yes 2) 17.5 3) 24 m 4) 16:1
Quick recap
- Similar figures: equal angles, proportional sides.
- Prove triangles similar with AA similarity, SSS or SAS.
- Use similar triangles for indirect measurement.