Class 8 Maths Part 2 Chapter 2: The Baudhayana-Pythagoras Theorem Notes

What is this chapter about?

Around 800 BCE, Baudhayana wrote a Shulbasutra, a manual for building fire altars of exact shapes and sizes. Builders needed to make one square twice as big as another, or one square equal to two others put together. Following his questions, the chapter discovers one of the most important results in geometry: the theorem that links the three sides of every right-angled triangle.

Doubling and halving a square

Doubling the side of a square does not double its area; it makes it four times as big. Baudhayana’s answer is to draw a square on the diagonal. North-south and east-west lines cut the original square into 2 identical right triangles and the new square into 4, so the new one has double the area. Running the idea backwards halves a square: fold the corners of a square paper to its centre, and the creases through the midpoints form a square with half the area.

The number √2

A right triangle whose two shorter sides are 1 unit is half of a unit square, and the square on its hypotenuse is the doubled square, with area 2. So the hypotenuse c satisfies c² = 2, and c = √2. More generally, equal sides a give c² = 2a², or c = a√2. Squaring decimals traps √2 ever more tightly: 1.4 < √2 < 1.5, then 1.41 < √2 < 1.42, then 1.414 < √2 < 1.415. Its decimal never ends, and, as Euclid proved around 300 BCE, it cannot be written as a fraction. Read the square root of 2.

Combining two squares: the theorem

What if two different squares must be combined into one? Baudhayana’s rule: make a right triangle whose shorter sides are the sides of the two squares; the square on its hypotenuse has exactly their total area. Putting the two squares side by side and moving two triangles proves it, which gives Baudhayana’s theorem: in any right triangle with shorter sides a and b and hypotenuse c, a² + b² = c². So a right triangle with sides 3 cm and 4 cm has a hypotenuse of exactly 5 cm, because 9 + 16 = 25. The theorem is also called the Pythagorean theorem, after Pythagoras, who studied it about three centuries after Baudhayana. Read the Baudhayana-Pythagoras theorem.

Baudhayana triples

Counting numbers (a, b, c) with a² + b² = c² are called Baudhayana triples or Pythagorean triples. Baudhayana himself listed six of them, including (3, 4, 5), (5, 12, 13), (7, 24, 25) and (8, 15, 17). Multiplying a triple by any counting number gives another, such as (6, 8, 10), so there are infinitely many. Triples with no common factor, like (3, 4, 5), are primitive. A neat trick finds more: the first n odd numbers add up to n², so (n − 1)² + (2n − 1) = n², and whenever 2n − 1 is itself a square, a new triple appears. Read Pythagorean triples.

A problem that took over 300 years

Since many squares are sums of two squares, the French mathematician Fermat asked in the 17th century whether a cube can be the sum of two cubes, or a fourth power the sum of two fourth powers. He wrote in a margin that xⁿ + yⁿ = zⁿ has no solution for n greater than 2, and that he had a proof the margin was too small to hold. Nobody found it for over 300 years. In 1963 a 10-year-old, Andrew Wiles, read about the problem and decided to solve it; he proved Fermat’s Last Theorem in 1994.

Using the theorem

The theorem finds lengths that cannot be measured directly. In Bhaskaracharya’s Lilavati, a lotus stands 1 unit above a lake and, blown by the breeze, touches the water 3 units away. If the depth is x, the stem is x + 1, and 3² + x² = (x + 1)², which gives x = 4. The same idea gives the diagonal of a square, the side of a rhombus from its diagonals, and the height of an equilateral triangle.

In a nutshell

For any right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides. Use a² + b² = c² to find a missing side, check whether three lengths make a right triangle, and build triples. The chapter ends with a puzzle about three wrongly labelled boxes. Answers to every question are in the NCERT solutions for Part 2 Chapter 2.