The Baudhayana–Pythagoras Theorem
This chapter explores one rule that quietly underpins a huge amount of geometry: the relationship binding the three sides of a right-angled triangle. We open with a builder’s challenge from almost three thousand years ago, can you lay out a square holding exactly twice the area of another?, and follow it patiently until it delivers the celebrated equation a^2 + b^2 = c^2. On the way we meet the diagonal of a square, the restless number \sqrt{2} that refuses to become a fraction, whole-number trios like (3,4,5) that snap perfectly onto a right triangle, a bamboo shoot bending over a pond, and a margin note that taunted mathematicians for three centuries. Our central figure is Baudhayana, who set down this theorem in its complete, modern shape long before Pythagoras lived.
Learning Outcomes
By the end of this chapter, you will be able to:
- build a square with twice (or half) the area of a given square by using its diagonal;
- compute the hypotenuse of an isosceles right triangle and show why it measures a\sqrt{2};
- explain why \sqrt{2} is neither a terminating decimal nor a fraction, and pin it between tight bounds;
- state and apply the Baudhayana–Pythagoras Theorem a^2+b^2=c^2 to recover any missing side of a right triangle;
- spot, scale, and generate Baudhayana (Pythagorean) triples, and separate primitive ones from scaled copies;
- put the theorem to work on real problems, diagonals, rhombuses, distances on a grid, the depth of a pond, and trace its history from the Sulba-Sutra through Fermat to Andrew Wiles.