ICSE Class 10 Maths: Similarity — Complete Chapter Notes 2026

Concept of Similarity

• Two figures are similar if same shape but different size; all corresponding angles equal; corresponding sides proportional
• Symbol: ~ (similar to); △ABC ~ △DEF means vertices correspond in that order
• Contrast with congruence: congruent figures same shape AND size (≅); similar figures only same shape

Criteria for Similarity of Triangles

• AA (Angle-Angle): two pairs of corresponding angles equal → triangles similar; sufficient because third angle determined
• SAS (Side-Angle-Side): ratio of two corresponding sides equal AND included angle equal
• SSS (Side-Side-Side): all three pairs of corresponding sides in same ratio

Basic Proportionality Theorem (BPT)

• If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio
• BPT: PQ || BC in △ABC → AP/PB = AQ/QC (Thales' theorem)
• Converse: if a line divides two sides of a triangle in same ratio, it is parallel to the third side

Applications of Similarity

• Heights and distances: similar triangles formed by shadow and pole; or lamp and observer
• Maps and scale drawing: map scale = actual distance × scale factor; similar figure application
• Building architecture: scale models similar to actual buildings; ratio calculations

Ratio of Areas and Perimeters

• Similar triangles with ratio k:1: ratio of perimeters = k:1; ratio of areas = k²:1
• Example: if sides in ratio 3:2, then areas in ratio 9:4
• ICSE frequently asks: given ratio of sides, find ratio of areas; or given ratio of areas, find ratio of sides

Pythagoras Theorem and Similarity

• Pythagoras: in right triangle, square on hypotenuse = sum of squares on other two sides; proved using similarity
• Converse: if a² + b² = c², the triangle has a right angle
• ICSE applications: heights of buildings, diagonal of rectangle, ladder problems, coordinate distance

ICSE Exam Strategy

• Similarity: 4–6 marks; proof of BPT and applications; ratio of areas most commonly tested
• Always show steps: identify similar triangles, state criteria, write proportion, solve
• Proof questions: learn BPT proof and Pythagoras proof; ICSE may ask theorem proof (4 marks)