• Identify and explain the Baudhäyana-Pythagoras Theorem.
• Calculate the area of an equilateral triangle using the altitude.
• Solve problems involving isosceles right triangles and their hypotenuses.
• Generate Baudhäyana triples using the sum of odd numbers.
• Apply Fermat's Last Theorem to integer solutions of equations.
• Construct geometric figures based on given properties and relationships.

Notes on the Baudhâyana-Pythagoras Theorem

Overview

• The Baudhâyana-Pythagoras Theorem is a fundamental theorem in geometry.
• It expresses the relationship among the three sides of a right-angled triangle.

Key Concepts

• If a, b, c are the sidelengths of a right-angled triangle, where c is the length of the hypotenuse, then:
  • Formula: a² + b² = c²
• In an isosceles triangle with sidelengths a, a, c, we have:
  • Relation: a² + a² = c², i.e., c = a√2

Important Values

• The number √2 lies between 1.414 and 1.415.
• It cannot be expressed as a terminating decimal or as a fraction of two positive integers.

Baudhäyana-Pythagoras Triples

• A triple (a, b, c) of positive integers satisfying a² + b² = c² is called a Baudhäyana-Pythagoras triple.
• Examples include:
  • (3, 4, 5)
  • (6, 8, 10)
  • (5, 12, 13)
• Infinitely many such triples can be constructed.

Additional Theorem

• The equation aⁿ + bⁿ = cⁿ has no solution in positive integers when n > 2. This is known as 'Fermat's Last Theorem', proven by Andrew Wiles in 1994.

Example Problems

1. Find the hypotenuse of an isosceles right triangle whose equal sides have length 12.
   • Given a = 12, using the formula: c² = 2 × 12² = 288.
   • Therefore, c is between 16 and 17 units.
2. If the hypotenuse of an isosceles right triangle is √72, find its other two sides.
   • Given c = √72, using the formula: c² = 2a².

Diagram Descriptions

• Geometric Construction: Diagrams illustrate properties of right triangles and squares, focusing on dimensions and partitioning.
• Right Triangle: A right triangle with legs labeled 27 and 45, indicating the right angle.
• Square Division: A square divided into four equal right triangles, demonstrating symmetry and congruence.

Conclusion

• The Baudhâyana-Pythagoras Theorem is essential for understanding the relationships in right triangles and has numerous applications in geometry.