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The Baudhayana - Pythagoras Theorem

Learning Objectives

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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.

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Revision Notes & Summary

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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

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.
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.

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Practice Test – MCQs, True/False

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