Similar triangles
How to prove two triangles similar with the right reason (AAA, 3 sides prop., ratio of 2 sides, inc. ∠) and use ratios of sides, areas and volumes. It is used in geometry proofs and in similar solids.
What you will learn
- 22.1 Similar triangles: same shape, possibly different size.
- 22.2 Conditions:
- 22.3 Similar 2-D figures. A quadrilateral with proportional sides is not necessarily similar.
You need first: Geometry and data basics
Key terms
| English | 中文 |
|---|---|
| similar triangles | 相似三角形 |
| corresponding sides / angles | 對應邊／對應角 |
| ratio of sides | 邊的比 |
| scale factor | 比例因子 |
| equiangular | 等角 |
Standard steps
Prove similar
- Name the triangles in matching order (△ADE ~ △ABC).
- Give the equal angles or ratios, one per line, each with a reason.
- Finish with the condition: "∴ △ADE ~ △ABC (AAA)".
Find a length
- Write the matching sides as a ratio, same triangle on top: AD/AB = DE/BC = AE/AC.
- Put in the numbers and solve.
- Check: the larger triangle must give the larger length.
Area and perimeter
- Find the side ratio from one pair of matching sides.
- Perimeter ratio = side ratio.
- Area ratio = (side ratio)².
Why: area uses two lengths multiplied, so the scale factor is used twice.
Worked examples
Example 1: In △ABC, D is on AB and E is on AC, with DE // BC. AD = 4 cm, DB = 2 cm and DE = 6 cm. Find BC and the ratio of the areas of △ADE and △ABC.
∠ADE = ∠ABC (corr. ∠s, DE // BC)
∠AED = ∠ACB (corr. ∠s, DE // BC)
∠A is common
∴ △ADE ~ △ABC (AAA)
AB = 4 + 2 = 6
DE/BC = AD/AB
6/BC = 4/6
BC = 9 cm
area ratio = (4/6)² = 4 : 9Example 2: Prove that △PQR and △XYZ are similar, given PQ = 4, QR = 6, RP = 8, XY = 6, YZ = 9 and ZX = 12.
XY/PQ = 6/4 = 3/2
YZ/QR = 9/6 = 3/2
ZX/RP = 12/8 = 3/2
∴ △PQR ~ △XYZ (3 sides proportional)Shortest with shortest, longest with longest: that is how the matching order is found.
Example 3: Prove similar, then find a length: AC and BD meet at O, with OA = 4 cm, OC = 6 cm, OB = 6 cm, OD = 9 cm and AB = 5 cm. Find CD.
∠AOB = ∠COD (vert. opp. ∠s)
OA/OC = 4/6 = 2/3
OB/OD = 6/9 = 2/3
∴ △OAB ~ △OCD (ratio of 2 sides, inc. ∠)
AB/CD = OA/OC
5/CD = 2/3
CD = 7.5 cmExample 4: Area and perimeter of similar triangles: corresponding sides are 3 cm and 5 cm. The smaller triangle has perimeter 12 cm and area 18 cm². Find the perimeter and area of the larger one.
side ratio = 3 : 5
perimeter = 12 × 5/3 = 20 cm
area = 18 × (5/3)² = 18 × 25/9 = 50 cm²Common mistakes
- Using DB instead of AB: the matching side of AD is the whole side AB.
- Writing the triangles in non-matching order, so the ratios pair the wrong sides.
- Using the side ratio for areas.
- Using "AA" when the textbook and markers expect "AAA" (check the student's textbook).
How the DSE tests it
- A(1) proof: prove two triangles similar, then find a length, perimeter or area.
- Paper 2: find a length from parallel lines, or the area of part of a figure.
- Later: similar solids (volume ratio = scale factor³), and proofs in circle geometry.
Questions you might not ask in class
How do I know which side matches which?
Match by angles: the side opposite the red angle in one triangle matches the side opposite the red angle in the other.
Why is it AD/AB and not AD/DB?
AD is a side of the small triangle. Its partner must be a side of the big triangle, and that is AB, not the piece DB.
Why are areas squared?
Doubling every length makes a rectangle twice as long and twice as wide, so its area is 4 times as big.