CBSE Class 10 Maths: Triangles — Similarity & Pythagoras Notes 2026

Basic Proportionality Theorem (Thales' Theorem)

• If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio
• Converse: if a line divides two sides of a triangle in the same ratio, it is parallel to the third side
• Application: find unknown side length when a parallel line divides the triangle

Similarity of Triangles

• Two figures are similar if they have same shape but may differ in size
• Similar triangles: corresponding angles equal; corresponding sides in same ratio (scale factor)
• AA, SSS, SAS similarity criteria (AA sufficient since angle sum is 180°)

Criteria for Similarity

• AA (Angle-Angle): two corresponding angles equal → triangles similar
• SSS: ratios of corresponding sides equal → similar
• SAS: one pair of equal angles between proportional sides → similar

Areas of Similar Triangles

• Ratio of areas of similar triangles = square of ratio of corresponding sides
• If ΔABC ~ ΔPQR, then ar(ABC)/ar(PQR) = (AB/PQ)² = (BC/QR)²
• Application: find area of one triangle given area of other and scale factor

Pythagoras Theorem

• In right triangle: (hypotenuse)² = (base)² + (perpendicular)²; i.e., c² = a² + b²
• Proof: using similar triangles (CBSE standard proof)
• Converse: if c² = a² + b², then angle opposite c is 90°

Application of Pythagoras Theorem

• In equilateral triangle of side a: height = a√3/2
• Find diagonal of rectangle, distance in coordinate geometry, height of tower
• CBSE: three consecutive Pythagorean triplets type questions

CBSE Exam — Triangles

• Prove: if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally
• ΔABC ~ ΔDEF; area = 36:81; find ratio of corresponding sides
• In right triangle ABC, right angle at B; prove AB² + BC² = AC² using similar triangles proof