Triangles – Class 10 Maths Chapter 6: Similarity, BPT & Pythagoras Theorem
Class 10 Triangles notes: similar triangles, BPT, AAA-SAS-SSS similarity criteria, Pythagoras theorem with proofs for CBSE & ICSE boards.
Chapter 6: Triangles
6.1 Similar Figures
Two figures are similar if they have the same shape (but not necessarily the same size). For polygons, this means corresponding angles are equal and corresponding sides are proportional.
6.2 Basic Proportionality Theorem (BPT / Thales' Theorem)
If DE ∥ BC in △ABC, then: AD/DB = AE/EC
Converse of BPT
If a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.
6.3 Criteria for Similarity of Triangles
1. AA (Angle-Angle) Similarity
If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
2. SSS (Side-Side-Side) Similarity
If the corresponding sides of two triangles are in the same ratio (proportional), the triangles are similar.
3. SAS (Side-Angle-Side) Similarity
If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the triangles are similar.
• Corresponding angles are equal: ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
• Corresponding sides are proportional: AB/DE = BC/EF = CA/FD
• Ratio of areas = (ratio of corresponding sides)² = (AB/DE)²
• Ratio of perimeters = ratio of corresponding sides
6.4 Pythagoras Theorem
AC² = AB² + BC² (where AC is the hypotenuse)
Converse of Pythagoras Theorem
If in a triangle, the square of one side equals the sum of the squares of the other two sides, then the angle opposite to the first side is a right angle.
Example: A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from the wall.
Let the distance = x. By Pythagoras: 10² = 8² + x² → 100 = 64 + x² → x² = 36 → x = 6 m
Indexed Topics & Examination Keywords
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