Squares and Square Roots

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

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

Halving a Square Using Paper

Cut out a square from a piece of paper. Now make a square whose area is half the area of the first square.

Will the square having half the sidelength have half the area? Why not? How many such squares will fill the original square?

Fold the square paper inward, as shown, such that the crease lines pass through the midpoints of the sides. PQRS is the required square with half the area.

There are 3 closed boxes—one containing only red balls, the second containing only blue balls and the third containing only green balls. The boxes are labelled RED, BLUE and GREEN such that ‘no’ box has the correct label. We need to find which label goes with which box. How can this be done if we are allowed to open only one box?

Earlier, we saw a method to create a square with double the area of a given square paper. There is another method to do this in which two identical square papers are cut in the following way.

Can you arrange these pieces to create a square with double the area of either square?

The length of the two equal sides of an isosceles right triangle is given. Find the length of the hypotenuse. Find bounds on the length of the hypotenuse such that they have at least one digit after the decimal point.

(i) 3

(ii) 4

(iii) 6

(iv) 8

(v) 9

The hypotenuse of an isosceles right triangle is 10. What are its other two sidelengths? [Hint: Find the area of the square composed of two such right triangles.]

If a right-angled triangle has shorter sides of lengths 5 cm and 12 cm, then what is the length of its hypotenuse? First draw the right-angled triangle with these sidelengths and measure the hypotenuse, then check your answer using Baudhāyana's Theorem.

If a right-angled triangle has a short side of length 8 cm and hypotenuse of length 17 cm, what is the length of the third side? Again, try drawing the triangle and measuring, and then check your answer using Baudhāyana's Theorem.

Using the constructions you have now seen, how would you construct a square whose area is triple the area of a given square? Five times the area of a given square? (Baudhāyana's Śulba-Sūtra, Verse 1.10)

Let $a$, $b$ and $c$ denote the length of the sides of a right triangle, with $c$ being the length of the hypotenuse. Find the missing sidelength in each of the following cases:

(i) $a = 5, b = 7$

(ii) $a = 8, b = 12$

(iii) $a = 9, c = 15$

(iv) $a = 7, b = 12$

(v) $a = 1.5, b = 3.5$

Find 5 more Baudhāyana triples using this idea.

Does this method yield non-primitive Baudhāyana triples?

[Hint: Observe that among the triples generated, one of the smaller sidelengths is one less than the hypotenuse.]

Are there primitive triples that cannot be obtained through this method? If yes, give examples.

Find the diagonal of a square with sidelength 5 cm.

Find the missing sidelengths in the following right triangles:

Find the sidelength of a rhombus whose diagonals are of length 24 units and 70 units.

Is the hypotenuse the longest side of a right triangle? Justify your answer.

True or False—Every Baudhāyana triple is either a primitive triple or a scaled version of a primitive triple.

Give 5 examples of rectangles whose sidelengths and diagonals are all integers.

Construct a square whose area is equal to the difference of the areas of squares of sidelengths 5 units and 7 units.

(i) Using the dots of a grid as the vertices, can you create a square that has an area of (a) 2 sq. units, (b) 3 sq. units, (c) 4 sq.units, and (d) 5 sq. unit?

(ii) Suppose the grid extends indefinitely. What are the possible integer-valued areas of squares you can create in this manner?

Find the area of an equilateral triangle with sidelength 6 units. [Hint: Show that an altitude bisects the opposite side. Use this to find the height.]

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

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

In many of the constructions in the Śulba-Sūtra, it is desirable to construct, where needed, what Baudhāyana 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.

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.**]

Context: 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.

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

Now suppose we are given a square, and we want to construct a square whose area is half that of the original square. How would you do it?

Why is the smaller inside square half the area of the larger square?

Again, adding some east-west and north-south lines can explain it:

Why is PQRS a square? Why is its area half that of the original paper?

Explain by connecting QS and PR, finding the different angles formed, and then using tringle congruence.

Find the hypotenuse of this isosceles right triangle.

What is the value of $\sqrt{2}$?

Is $\sqrt{2}$ less than or greater than 1?

Is $\sqrt{2}$ less than or greater than 2?

Can we find closer bounds for $\sqrt{2}$?

Will we ever get a number with a terminating decimal representation whose square is 2?

If there is such a terminating decimal starting with 1.414... whose square is 2, then it must have a non-zero last digit. If this is the case, then the decimal representation of its square will also have a non-zero last digit after the decimal point. For example, if $\sqrt{2}$ is of the form 1.414...4, then its square will be of the form—

Use this formula to check your answers in the Figure it Out on page 39.

What if we wish to combine two squares of 'different' sizes to make a large square whose area is the sum of the two smaller squares?

Why does Baudhāyana’s method work?

Can you see why the method works in the case where the two squares are the same size? Does it agree with the method we used earlier to combine two same sized squares into a bigger square?

Explain why all the angles of this new 4-sided figure are right angles and so it is a square.

List down all the Baudhāyana triples with numbers less than or equal to 20.

Is there an unending sequence of Baudhāyana triples?

Is (30, 40, 50) a Baudhāyana triple?

Is (300, 400, 500) a Baudhāyana triple?

Context: The list of Baudhāyana triples having numbers less than or equal to 20 contains the following triples — (3, 4, 5), (6, 8, 10), (9, 12, 15), (12, 16, 20).

Q. Do you see any pattern among them?

Context: All these triples can be obtained by multiplying each term of (3, 4, 5) by a certain positive integer.

Q. Can we form a conjecture on Baudhāyana triples based on this observation?

Context: Conjecture: (3k, 4k, 5k) is a Baudhāyana triple, where k is any positive integer.

Q. Is this true?

Is $(5, 12, 13)$ a primitive Baudhayana triple? What are the other primitive Baudhayana triples with numbers less than or equal to 20?

Generate 5 scaled versions of each of these primitive triples. Are these scaled versions primitive?

If $(a, b, c)$ is non-primitive, and the integers have $f$ — greater than 1 — as a common factor, then is $\left(\frac{a}{f}, \frac{b}{f}, \frac{c}{f}\right)$ a Baudhayana triple? Check this statement for $(9, 12, 15)$. Justify this statement.