Squares and Square Roots | IT

Question 4

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?

Solution

Understand the Question

The original square of side length $s$ is divided by a diagonal into $2$ right-angled isosceles triangles.
The new square has double the area of the original square, so its side length is $s\sqrt{2}$ . When divided by both its diagonals, it forms $4$ smaller triangles.
To explain why all these triangles are congruent, we determine the side lengths and angles of both sets of triangles and show that they are identical.

Step 1 · Analyze Triangles in the Original Square

Diagram 1

Let the side length of the original square be $s$ .

A diagonal divides the square into two right-angled triangles with two perpendicular sides of length $s$ and an included angle of $90^\circ$ .

By Pythagoras theorem:

\begin{aligned} \text{Hypotenuse}^2 &= s^2 + s^2 \\ &= 2s^2 \\ \text{Hypotenuse} &= s\sqrt{2} \end{aligned}

Thus, each of the $2$ triangles in the original square has sides of length $s, s, s\sqrt{2}$ and angles $90^\circ, 45^\circ, 45^\circ$ .

Step 2 · Analyze Triangles in the New Square

Diagram 2

The area of the new square is double that of the original square:

$\text{Area} = 2s^2$

Let $S$ be the side length of the new square:

\begin{aligned} S^2 &= 2s^2 \\ S &= s\sqrt{2} \end{aligned}

The diagonals of a square bisect each other perpendicularly at $90^\circ$ . The length of each diagonal is:

\begin{aligned} \text{Diagonal length} &= (s\sqrt{2}) \times \sqrt{2} \\ &= 2s \end{aligned}

Each of the $4$ triangles formed by the intersecting diagonals has:

Two legs equal to half the diagonal length: $\dfrac{2s}{2} = s$
Included angle between legs: $90^\circ$
Hypotenuse equal to the side of the new square: $S = s\sqrt{2}$

Thus, each of the $4$ triangles in the new square also has sides of length $s, s, s\sqrt{2}$ and angles $90^\circ, 45^\circ, 45^\circ$ .

Step 3 · Establish Congruence

Comparing the triangles from both squares:

Both the $2$ triangles from the original square and the $4$ triangles from the new square are right-angled isosceles triangles.
All of them have identical side lengths: $s$ , $s$ , and $s\sqrt{2}$ .

Therefore, by the SSS (or SAS) congruence criterion, all the small triangles are congruent to each other.

Answer

All the small triangles are congruent because they are all right-angled isosceles triangles with identical side lengths $s, s, s\sqrt{2}$ (by SSS or SAS congruence criterion).

Common Mistakes

Doubling Sides Instead of Area: Doubling the area of a square multiplies its side length by $\sqrt{2}$ , not by $2$ .
Overlooking Diagonal Properties: Forgetting that the diagonals of a square are perpendicular bisectors of each other, which guarantees that the legs of the $4$ small triangles in the new square are $\dfrac{2s}{2} = s$ at right angles ( $90^\circ$ ).

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?

[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 $\sqrt{2}$ ?

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 $1.414\dots$ 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 $1.414\dots4$ , 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 $\left(\dfrac{a}{f}, \dfrac{b}{f}, \dfrac{c}{f}\right)$ a Baudhayana triple? Check this statement for $(9, 12, 15)$ . Justify this statement.