Chapter 9: The Baudhayana-Pythagoras Theorem

In a right-angled triangle, the longest side , the hypotenuse , has a magical relationship with the other two sides. If you build a square on each side, the square on the hypotenuse has area equal to the sum of the squares on the other two. In symbols:

$a^2 + b^2 = c^2,$

where $c$ is the hypotenuse. This is the Pythagoras theorem , but it was known to the Indian mathematician Baudhayana at least $2{,}500$ years ago, written in the Baudhayana Sulba Sutra well before the Greek Pythagoras was born. So in India this result is correctly called the Baudhayana-Pythagoras theorem.

The theorem matters because right triangles are everywhere , in building corners, in computer graphics, in maps, in physics. Whenever a problem involves perpendicular distances, the Baudhayana-Pythagoras theorem turns up.

In this chapter you will learn what the theorem says, see it proved (in more than one way), use it to compute unknown sides, meet the special integer triples like $(3,4,5)$ that satisfy it, and apply it in everyday situations.

By the end you will be able to recognise the theorem in any problem that hides a right angle, and you will use it as confidently as you use multiplication.

What's inside

1. The theorem and a visual proof , Sulba Sutra origins and a rearrangement argument.
2. Computing the missing side , given any two, find the third.
3. Pythagorean triples , special integer right triangles.
4. Applications , ladders, ramps, screen sizes, distances.
5. Converse and a touch of trigonometry , when is a triangle right-angled?

Key results

Common Pythagorean triples: $(3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (9, 40, 41)$. Multiples of any triple are also triples , $(6, 8, 10), (9, 12, 15), \dots$

How to read this chapter

Draw a right triangle on graph paper, mark the squares on each side, and count the unit squares inside each. You will literally see the theorem before you read its proof.