NCERT Class 8 Mathematics Ganita Prakash Part 2 Chapter 02 The Baudhayana Pythagoras Theorem PDF Download

Chapter 2: The Baudhayana-Pythagoras Theorem

2.1 Doubling a Square

In Baudhayana's Sulba-Sutra (c. 800 BCE), Baudhayana considers the following question:

How can one construct a square having double the area of a given square?

A first guess might be to simply double the length of each side of the square. Will this new square have double the area of the original square?

It's not hard to see that the new square will have area \( 2 \times 2 = 4 \) times the area of the original square.

So how can one make a square that has double the area? Baudhayana in his Sulba-Sutra (Verse 1.9) gave an elegant answer to this question:

The diagonal of a square produces a square of double the area of the original square.

As Baudhayana says, the key is to construct a square on the diagonal of the original square:

[Figure: Original square with a rotated square constructed on its diagonal, See in your textbook]

Teacher's Note

When you double a square by simply making each side twice as long, you don't get double the area - you get 4 times the area. This is because both length and width increase, so the effect multiplies. The clever insight here is that the diagonal creates a different square that has exactly twice the area, not four times.

Math Talk

Why does the new dotted square have double the area of the original square?

In many of the constructions in the Sulba-Sutra, it is desirable to construct, where needed, what Baudhayana calls 'east-west' and 'north-south' lines, i.e., horizontal and vertical lines that are perpendicular to each other. Can you draw some horizontal and vertical lines to see why the new square has double the area of the original square?

You could draw some horizontal and vertical lines as shown on the right.

[Figure: Square with diagonal and horizontal/vertical lines through the center, See in your textbook]

Why should the extension of the vertical and horizontal sides of the original square pass through the vertices of the dotted square?

Hint: From the diagonal property of a square, the line that bisects an angle passes through the opposite vertex. Argue why the vertical and horizontal sides of the original square bisect the two angles of the dotted square.

So, the new square has double the area of the original square, because the original square is made up of two small triangles, while the new square is made up of four small triangles.

Moreover, all these small triangles are congruent to each other. Can you explain why?

Adding some more horizontal and vertical lines can make the situation even clearer:

[Figure: Two nested squares with horizontal and vertical grid lines showing eight congruent triangles, See in your textbook]

Teacher's Note

The reason all eight triangles are congruent is that they are created by perpendicular lines passing through the center of both squares. Each triangle has the same base and height, so they have equal area and the same shape. This is why counting triangles - 2 in the small square, 4 in the large square - proves the area exactly doubles.

So we can make the following sequence of squares:

[Figure: Three diagrams showing a square subdivided into 2 triangles, the rotated square with 4 triangles, and the next square with 8 triangles, See in your textbook]

Each square has double the area of the previous one, as they are made up of 2, 4, and 8 small triangles, respectively.

Doubling a Square Using Paper

Cut out two identical squares of paper. Draw, label, and cut as follows:

[Figure: Square 1 showing four triangles labeled 1, 2, 3, 4 from the two diagonals, See in your textbook]

[Figure: Identical Square 2 showing four corner triangles labeled 5, 6, 7, 8 and a central square, See in your textbook]

Now place the pieces 5, 6, 7, and 8 around Square 1 to get a square with double the area.

Key Points

• Doubling the side length of a square gives an area 4 times larger, not 2 times larger, because both dimensions increase.
• A square constructed on the diagonal of an original square has exactly double the area of that original square.
• The doubling works because the new square contains four congruent triangles while the original contains two congruent triangles of the same size and shape.
• This construction from Baudhayana's Sulba-Sutra demonstrates a geometric principle that is connected to what is later known as the Pythagoras theorem.

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