What is the difference between similar and congruent triangles?

Congruent triangles have the same shape AND same size (all corresponding sides and angles equal). Similar triangles have the same shape but may differ in size (corresponding angles equal, corresponding sides proportional). Congruence (≅) is a special case of similarity (~). All congruent triangles are similar, but not all similar triangles are congruent.

State and explain the Basic Proportionality Theorem (BPT).

BPT (Thales' Theorem): If a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides in the same ratio. If DE ∥ BC in △ABC, then AD/DB = AE/EC. Converse: If AD/DB = AE/EC, then DE ∥ BC.

What are Pythagorean triples and how are they useful?

Pythagorean triples are sets of three whole numbers (a, b, c) satisfying a² + b² = c². Common ones: (3,4,5), (5,12,13), (8,15,17), (7,24,25). They are useful because: (1) Quick identification of right triangles without calculation, (2) Used in construction problems, (3) Multiples are also triples (e.g., 6,8,10 or 9,12,15).

Similarity of Triangles

Triangles • Class 10 Mathematics • NCERT • CBSE

Two triangles are similar if they have the same shape but not necessarily the same size. The criteria for similarity are AA (Angle-Angle), SSS (Side-Side-Side ratio), and SAS (Side-Angle-Side ratio). The Basic Proportionality Theorem (BPT) is a fundamental result.

Key Formulas

BPT: AD/DB = AE/EC (when DE ∥ BC in △ABC)
Area ratio of similar △ = square of ratio of corresponding sides
Pythagoras: c² = a² + b² (right triangle)
Common triples: (3,4,5), (5,12,13), (8,15,17)

Frequently Asked Questions

What is the difference between similar and congruent triangles?: Congruent triangles have the same shape AND same size (all corresponding sides and angles equal). Similar triangles have the same shape but may differ in size (corresponding angles equal, corresponding sides proportional). Congruence (≅) is a special case of similarity (~). All congruent triangles are similar, but not all similar triangles are congruent.
State and explain the Basic Proportionality Theorem (BPT).: BPT (Thales' Theorem): If a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides in the same ratio. If DE ∥ BC in △ABC, then AD/DB = AE/EC. Converse: If AD/DB = AE/EC, then DE ∥ BC.
What are Pythagorean triples and how are they useful?: Pythagorean triples are sets of three whole numbers (a, b, c) satisfying a² + b² = c². Common ones: (3,4,5), (5,12,13), (8,15,17), (7,24,25). They are useful because: (1) Quick identification of right triangles without calculation, (2) Used in construction problems, (3) Multiples are also triples (e.g., 6,8,10 or 9,12,15).

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