Squares and Square Roots | IT

Question 3

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.**]

Solution

The extensions of the solid square's sides form the axes, which are the diagonals of the dotted square.

Step 1 — Setting up the coordinate system

Let us place the common vertex of the solid square and the center of the dotted square at the origin, which is point $O(0,0)$ . The solid square has its sides perfectly horizontal and vertical. So, the line extending from its horizontal side is the x-axis. The line extending from its vertical side is the y-axis.

Diagram 1

Step 2 — Understanding angle bisectors in a square

Let the dotted square be $ABCD$ . Its center is $O$ . Each corner (vertex) of a square has an angle of $90^\circ$ . A special property of a square is that its diagonals bisect these vertex angles. For example, the diagonal $AC$ bisects the angle at vertex $A$ and also the angle at vertex $C$ . Similarly, the diagonal $BD$ bisects the angle at vertex $B$ and the angle at vertex $D$ .

Step 3 — Connecting the extensions to the diagonals

We know the x-axis and y-axis are perpendicular lines. They pass through the center $O$ of the dotted square. The hint asks us to argue that the x-axis and y-axis (extensions of the solid square's sides) bisect two angles of the dotted square. If the x-axis bisects an angle of the dotted square, it must be one of its diagonals. If the y-axis bisects another angle of the dotted square, it must be the other diagonal. Since the diagonals of a square are perpendicular and intersect at its center, it makes perfect sense for the x-axis and y-axis to be the diagonals of the dotted square.

Step 4 — Concluding the position of the vertices

If the x-axis is a diagonal of the dotted square, then two of its vertices must lie on the x-axis. If the y-axis is the other diagonal of the dotted square, then the other two vertices must lie on the y-axis. This means all four vertices of the dotted square are located on either the x-axis or the y-axis. Since the x-axis is the extension of the horizontal side of the original square, it passes through two vertices of the dotted square. Since the y-axis is the extension of the vertical side of the original square, it passes through the other two vertices of the dotted square. Therefore, the extensions of the vertical and horizontal sides of the original square pass through the vertices of the dotted square.

Answer

The common vertex of the solid square is the center of the dotted square. The extensions of the solid square's horizontal and vertical sides form two perpendicular lines (the x-axis and y-axis). These lines pass through the center of the dotted square. For these lines to bisect the vertex angles of the dotted square (as suggested by the hint), they must be the diagonals of the dotted square. Since the vertices of a square lie on its diagonals, the extensions of the original square's sides must pass through the vertices of the dotted square.

More questions in IT

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?

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 $\sqrt{2}$ ?

Q10

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

Q11

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

Q12

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

Q13

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—

Q14

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

Q15

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?

Q16

Why does Baudhāyana’s method work?

Q17

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?

Q19

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

Q20

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

Q21

Is there an unending sequence of Baudhāyana triples?

Q22

Is (30, 40, 50) a Baudhāyana triple?

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

Q23

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?

Q24

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?

Q25

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

Q. Is this true?

Q26

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

Q27

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

Q28

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.