Squares and Square Roots | FIO

Question 10

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

Solution

Understand the Question

The formula $\left(x, \dfrac{x^2-1}{2}, \dfrac{x^2+1}{2}\right)$ generates Pythagorean triples for any odd integer $x \ge 3$ .
In all triples generated by this method, the smallest side length is always the odd integer $x$ , and the difference between the hypotenuse and the larger leg is always $1$ .
A primitive Pythagorean triple $(a, b, c)$ is one where $\gcd(a, b, c) = 1$ .
Any primitive Pythagorean triple where the smallest leg is even (or where the difference between the hypotenuse and the largest leg is not $1$ ) cannot be generated by this method.

Step 1 · Analyze the Given Triple Formula

For any odd integer $x$ :

x^2 + \left(\dfrac{x^2-1}{2}\right)^2 = \left(\dfrac{x^2+1}{2}\right)^2

For $x = 3$ :

\left(3, \dfrac{3^2-1}{2}, \dfrac{3^2+1}{2}\right) = (3, 4, 5)

For $x = 5$ :

\left(5, \dfrac{5^2-1}{2}, \dfrac{5^2+1}{2}\right) = (5, 12, 13)

For any odd number $x \ge 3$ , the smallest number in the generated triple is always the odd integer $x$ .

Step 2 · Find Primitive Triples Not Generated by the Method

A primitive Pythagorean triple has $\gcd(a, b, c) = 1$ .

Because this method only generates triples where the smallest side is odd, any primitive triple where the smallest number is even cannot be obtained:

$(8, 15, 17)$ : $8^2 + 15^2 = 64 + 225 = 289 = 17^2$ Here, $\gcd(8, 15, 17) = 1$ , and the smallest number is $8$ (even).
$(12, 35, 37)$ : $12^2 + 35^2 = 144 + 1225 = 1369 = 37^2$ Here, $\gcd(12, 35, 37) = 1$ , and the smallest number is $12$ (even).
$(16, 63, 65)$ : $16^2 + 63^2 = 256 + 3969 = 4225 = 65^2$ Here, $\gcd(16, 63, 65) = 1$ , and the smallest number is $16$ (even).

Answer

Yes, examples of primitive triples that cannot be obtained through this method include $(8, 15, 17)$ , $(12, 35, 37)$ , and $(16, 63, 65)$ .

Common Mistakes

Assuming All Primitive Triples Have an Odd Smallest Leg: In primitive triples like $(8, 15, 17)$ and $(20, 21, 29)$ , the even leg is smaller than the odd leg.
Confusing Primitive with Non-Primitive Triples: Triples like $(6, 8, 10)$ have an even smallest number but are not primitive because $\gcd(6, 8, 10) = 2$ .

More questions in FIO

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 $5\text{ cm}$ and $12\text{ cm}$ , 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.]

Q10

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

Q11

Find the diagonal of a square with sidelength $5 \text{ cm}$ .

Q12

Find the missing sidelengths in the following right triangles:

Q13

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

Q14

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

Q15

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

Q16

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

Q17

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

Q18

(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. units?

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

Q19

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