Triangles
Similar triangles, Basic Proportionality Theorem, and Pythagoras applications.
Chapter Overview
Class 10 Triangles extends congruence to similarity. You study criteria for similar triangles, Basic Proportionality Theorem (Thales), Pythagoras theorem and its converse, and how areas of similar triangles relate to the ratio of corresponding sides.
Similarity Criteria
- AAA: all corresponding angles equal
- SSS: sides in proportion
- SAS: one angle equal and including sides in proportion
- Similar figures have same shape but not necessarily same size
Key Theorems
BPT (Thales)
If DE ∥ BC in ΔABC, then AD/DB = AE/EC
Pythagoras
In right Δ: (hypotenuse)² = (base)² + (perpendicular)²
Area ratio
Area(Δ₁)/Area(Δ₂) = (side₁/side₂)²
Applications
- Use similarity to find unknown lengths in nested triangles
- Converse of BPT helps prove lines parallel
- Pythagoras converse: if a² + b² = c², angle opposite c is 90°
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: BPT
In ΔABC, DE ∥ BC. AD = 4 cm, DB = 6 cm, AE = 5 cm. Find EC.
- 1By BPT: AD/DB = AE/EC
- 24/6 = 5/EC
- 3EC = 30/4 = 7.5 cm
Answer
EC = 7.5 cm
Example 2: Pythagoras
A right triangle has legs 9 cm and 12 cm. Find hypotenuse.
- 1h² = 9² + 12² = 81 + 144 = 225
- 2h = √225 = 15
Answer
Hypotenuse = 15 cm
Key Points to Remember
- ✓ Similar triangles have corresponding angles equal and sides proportional
- ✓ Ratio of areas = square of ratio of corresponding sides
- ✓ BPT is also called Thales' theorem
Exam Tips
- • State similarity criterion before proving triangles similar
- • Draw parallel lines clearly in BPT problems
- • Check which side is hypotenuse in Pythagoras questions
Practice MCQs — Triangles
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
If two triangles are similar, corresponding angles are: