Learning Objectives
- Understand and apply the Basic Proportionality Theorem (BPT / Thales' Theorem)
- Learn criteria for similarity of triangles (AA, SSS, SAS)
- Apply the Pythagoras Theorem and its converse
- Relate areas of similar triangles to the ratio of their sides
Key Concepts
Similar Triangles
Two triangles are similar if their corresponding angles are equal and corresponding sides are in the same ratio (proportional). Symbol: △ABC ~ △DEF.
Basic Proportionality Theorem (Thales' Theorem)
If a line is drawn parallel to one side of a triangle to intersect the other two sides, it divides those two sides in the same ratio.
If DE ∥ BC in △ABC, and D is on AB, E is on AC, then: AD/DB = AE/EC.
Converse: If a line divides two sides of a triangle in the same ratio, it is parallel to the third side.
Criteria for Similarity
- AA (Angle-Angle): If two angles of one triangle are equal to two angles of another, the triangles are similar.
- SSS (Side-Side-Side): If the corresponding sides of two triangles are in the same ratio, they are similar.
- SAS (Side-Angle-Side): If one angle is equal and the sides including that angle are in the same ratio, the triangles are similar.
Areas of Similar Triangles
The ratio of the areas of two similar triangles equals the square of the ratio of their corresponding sides:
Area(△ABC) / Area(△DEF) = (AB/DE)² = (BC/EF)² = (CA/FD)²
Pythagoras Theorem
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:
AC² = AB² + BC² (where ∠B = 90°)
Converse: If AC² = AB² + BC² in a triangle, then ∠B = 90°.
Summary
Similar triangles have equal corresponding angles and proportional sides. BPT relates parallel lines to proportional division of triangle sides. AA, SSS, and SAS are criteria for proving similarity. The Pythagoras Theorem is a fundamental property of right triangles. Areas of similar triangles are proportional to the square of corresponding sides.
Important Terms
- Similar Triangles
- Triangles with equal corresponding angles and proportional corresponding sides
- Congruent Triangles
- Triangles that are identical in shape and size (special case of similarity with ratio 1:1)
- Hypotenuse
- The side opposite to the right angle in a right triangle; the longest side
Quick Revision
- BPT: Line parallel to one side divides the other two sides proportionally
- AA similarity: Two equal angles are sufficient to prove similarity
- Area ratio = (side ratio)² for similar triangles
- Pythagoras: Hypotenuse² = Base² + Perpendicular²
- Converse of Pythagoras can be used to check if a triangle is right-angled