Thales Theorem and Basic Proportionality Theorem

Introduction

• Overview of the lecture on Thales theorem (Basic Proportionality Theorem).
• Mention of other courses available (Physics, Chemistry, Biology, Maths for CBSC Class 8-10).

What is Thales Theorem?

• Definition: If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.
• Diagram Explanation:
  • Triangle ABC with line DE parallel to side BC.
  • Intersects AB at D and AC at E.
  • The ratios are:
    • AD/DB = AE/EC
• Example with lengths:
  • AD = 2 cm, DB = 4 cm, AE = 3 cm, EC = 6 cm.
  • AD/DB = 2/4 = 1/2 and AE/EC = 3/6 = 1/2 (confirming the theorem).

Proof of Thales Theorem

• Intuitive Understanding:
  • Similar triangles formed, corresponding angles are equal.
  • Not comparing AD to AB but rather the segments created by DE.
• Area Approach to Proof:
  • Area formula: Area = 1/2 * Base * Height.
  • Draw perpendiculars from D and E to respective sides.
  • Establish ratio of areas of triangles to prove AD/DB = AE/EC.

Corollaries of Thales Theorem

• Ratios can be expressed in different forms:
  • AD/DB = AE/EC
  • AB/DB = AC/EC
  • AB/AD = AC/AE
• Proof of Corollaries:
  • Start with Thales theorem ratio and add one to both sides.

Converse of Thales Theorem

• Definition: If a line divides two sides of a triangle in the same ratio, then it is parallel to the third side.
• Proof by Contradiction:
  • Assume line DE is not parallel to BC.
  • Draw line DF parallel to BC and apply Thales theorem.
  • Show that this leads to a contradiction.

Practice Problems

• Example problems on applying Thales theorem to find unknown lengths in given triangles.
• Use of Thales theorem or similarity of triangles to solve for unknowns.

Conclusion

• Recap of Thales theorem, its proof, corollaries, and converse.
• Encouragement to practice problems and check out other courses offered.