Class 10 Maths Notes — Chapter 6: Triangles
Similarity of triangles, Basic Proportionality Theorem (Thales), and related area results.
Detailed NCERT notes
- Two triangles are similar if (a) corresponding angles are equal AND (b) corresponding sides are in the same ratio.
- Basic Proportionality Theorem (BPT / Thales): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
- Converse of BPT: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
- Similarity criteria: AA (two angles equal ⇒ third equal, so similar), SAS (proportional sides + included angle equal), SSS (three sides proportional).
- Ratio of areas of two similar triangles = ratio of squares of any pair of corresponding sides (or medians, or altitudes, or perimeters).
- In similar triangles, ratio of perimeters = ratio of corresponding sides.
- Standard configurations to spot: line parallel to one side; two triangles sharing an angle at vertex.
- Pythagoras Theorem (statement retained in rationalised syllabus): In a right triangle, square of hypotenuse = sum of squares of the other two sides.
- Converse of Pythagoras: if a² + b² = c², then the triangle is right-angled at the vertex opposite to c.
- Common competency setup: light-pole and shadow, ladder against a wall, tree broken by wind — reduce to similar right triangles.
Formulas & key results
- BPT: DE ∥ BC ⇒ AD/DB = AE/EC
- Similarity criteria: AA, SAS, SSS
- Ratio of areas of similar △ = (ratio of sides)²
- Pythagoras: a² + b² = c² (right-angled)
Mind map
- Similarity → BPT → criteria → areas
- Pythagoras & its converse
- Ratios: sides ↔ perimeters ↔ altitudes ↔ medians ↔ √areas
Tricks & shortcuts
- Look for parallels → BPT.
- Look for common angle → AA similarity.
- For area ratio problems: work with the square of side ratio.
Common mistakes to avoid
- Assuming similar ⇒ congruent.
- Wrong direction of ratio (AD/AB vs AD/DB).
- Squaring only one side of the ratio when comparing areas.
Competency-based questions & answers
- Q. In △ABC, DE ∥ BC with AD = 3, DB = 5, AE = 4.5. Find EC.A. AD/DB = AE/EC ⇒ 3/5 = 4.5/EC ⇒ EC = 7.5.
- Q. Areas of two similar triangles are 81 cm² and 49 cm². If the longest side of the smaller is 14 cm, find longest side of the larger.A. Ratio of areas 81:49 ⇒ ratio of sides 9:7 ⇒ longest side = 14 × 9/7 = 18 cm.
- Q. A vertical pole 6 m casts a shadow 4 m long. At the same time, a tower casts a shadow 28 m. Find the height of the tower.A. Similar triangles: h/28 = 6/4 ⇒ h = 42 m.