Questions
5 questions per paper
Difficulty
Medium
Importance
Core — never skip
Overview
Triangles is a foundational geometry topic focusing on the properties of similar triangles and right-angled relationships. Mastering this area is essential as it serves as the backbone for trigonometry, vectors, and coordinate geometry in competitive and board exams.
Similarity Criteria
Two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio. Identifying the correct correspondence between vertices is critical for setting up ratios accurately.
- AAA (Angle-Angle-Angle) similarity
- SAS (Side-Angle-Side) similarity
- SSS (Side-Side-Side) similarity
- Ratio of areas of similar triangles is the square of the ratio of corresponding sides
- Corresponding medians, altitudes, and bisectors follow the same ratio as sides
Basic Proportionality Theorem (BPT/Thales Theorem)
BPT states that a line drawn parallel to one side of a triangle intersecting the other two sides divides them in the same ratio. This is the most frequently tested theorem for solving segment length problems.
- If DE || BC, then AD/DB = AE/EC
- Converse of BPT: If a line divides two sides proportionally, it is parallel to the third
- Corollaries include AD/AB = AE/AC = DE/BC
- Essential for problems involving transversal lines in a triangle
Pythagoras Theorem
The Pythagoras theorem establishes the relationship between the sides of a right-angled triangle. Its application extends beyond basic geometry into 3D mensuration and coordinate distance formulas.
- Hypotenuse squared = Sum of squares of the other two sides (a² + b² = c²)
- In a right triangle, the altitude to the hypotenuse divides the triangle into two similar triangles
- Property: Area of a square on the hypotenuse is the sum of the areas of squares on the other two sides
- Common Pythagorean triples: (3, 4, 5), (5, 12, 13), (8, 15, 17)
Formula Sheet
Area1/Area2 = (s1/s2)²
AD/DB = AE/EC
a² + b² = c²
Exam Tip
Always draw a clean diagram and label vertices in sequential order (clockwise or counter-clockwise) to avoid correspondence errors in similarity proofs.
Common Mistakes
- Assuming similarity without verifying corresponding vertex order (e.g., writing Triangle ABC is similar to Triangle EDF when the mapping is different).
- Confusing the ratio of perimeters (which is linear) with the ratio of areas (which is squared) in similar triangles.
- Attempting to apply BPT to non-parallel lines or missing the converse condition.
More Revision Notes
Ready to test yourself?
Play topic-wise Triangles questions in Aspirant Arcade — gamified MCQ practice.
Download Free