CLASS 8 MATHEMATICS • STUDY NOTES

Baudhāyana Triples

Right-Angled Triangles Having Integer Side Lengths

Understand the Pythagorean theorem, the names of the sides of a right-angled triangle, primitive triples, scaled triples and solved examples.

The Busy Brains

What Are Baudhāyana Triples?

In his Sulba-Sutra, Baudhāyana described sets of three integers that can form the side lengths of a right-angled triangle. Such sets are called Baudhāyana triples. They are also known as Pythagorean triples.

a² + b² = c²

Here, a and b represent the two sides that form the right angle, while c is the hypotenuse.

Example: (3, 4, 5)

3² + 4² = 5²
9 + 16 = 25
25 = 25 ✓

Therefore, (3, 4, 5) is a Baudhāyana triple.

Example: (5, 12, 13)

5² + 12² = 13²
25 + 144 = 169
169 = 169 ✓

Therefore, (5, 12, 13) is also a Baudhāyana triple.

Remember: A Baudhāyana triple consists of three positive integers satisfying the Pythagorean theorem.

Names of the Sides of a Right-Angled Triangle

A right-angled triangle has one angle equal to 90°. Its sides are named according to their positions.

Illustration: a 3–4–5 right-angled triangle.

Hypotenuse

It is always the longest side.
It lies opposite the 90° angle.
In the diagram, AC = 5 cm.

Perpendicular

One of the two sides forming the right angle.
AB = 3 cm in the example.

Base

The other side forming the right angle.
BC = 4 cm in the example.

Important clarification: The hypotenuse is always the longest side. The base and perpendicular are the two sides forming the right angle; neither one must always be the shortest side. Their names depend on the chosen orientation and reference.

The Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

a² + b² = c²

When the hypotenuse is unknown

c² = a² + b²
c = √(a² + b²)

When one shorter side is unknown

a² = c² − b²
a = √(c² − b²)

The theorem applies to right-angled triangles. Always identify the hypotenuse correctly before substituting values.

Solved Example: Finding the Hypotenuse

Question 1

A right-angled triangle has shorter sides of 5 cm and 12 cm. Find its hypotenuse.

Let the hypotenuse be c.
Using the theorem: c² = a² + b².

Substitute the values.

c² = 5² + 12²

Calculate the squares.

c² = 25 + 144 = 169

Take the square root.

c = √169 = 13 cm

Final answer: The hypotenuse is 13 cm.

Solved Example: Finding a Missing Side

Question 2

A right-angled triangle has a shorter side of 8 cm and a hypotenuse of 17 cm. Find the other shorter side.

Let the missing side be b.
8² + b² = 17²
64 + b² = 289
b² = 289 − 64
b² = 225
b = √225 = 15 cm

Final answer: The missing side is 15 cm.
The triple is (8, 15, 17).

Another example: a = 9, c = 15

9² + b² = 15²
81 + b² = 225
b² = 144
b = 12

Answer: b = 12. The triple is (9, 12, 15).

Important Baudhāyana Triples

Some commonly used triples are listed below. Each one satisfies a² + b² = c².

Triple	Verification	Primitive?
(3, 4, 5)	9 + 16 = 25	Yes
(5, 12, 13)	25 + 144 = 169	Yes
(8, 15, 17)	64 + 225 = 289	Yes
(7, 24, 25)	49 + 576 = 625	Yes
(12, 35, 37)	144 + 1225 = 1369	Yes
(6, 8, 10)	36 + 64 = 100	No
(9, 12, 15)	81 + 144 = 225	No

The textbook lists triples such as (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (12, 35, 37) and (15, 36, 39). Notice that (15, 36, 39) is a scaled version of (5, 12, 13).

Scaled Baudhāyana Triples

If (a, b, c) is a Baudhāyana triple, multiplying all three numbers by the same positive integer k gives another Baudhāyana triple.

(ka, kb, kc)

Example: Starting with (3, 4, 5)

Multiplier k	New triple	Check
1	(3, 4, 5)	9 + 16 = 25
2	(6, 8, 10)	36 + 64 = 100
3	(9, 12, 15)	81 + 144 = 225
4	(12, 16, 20)	144 + 256 = 400
5	(15, 20, 25)	225 + 400 = 625

Why does this work?

(ka)² + (kb)² = k²a² + k²b²
= k²(a² + b²)
= k²c²
= (kc)²

Therefore, (ka, kb, kc) is a Baudhāyana triple for every positive integer k. This gives infinitely many triples.

Primitive and Non-Primitive Triples

Primitive triple

A triple is primitive if its three numbers have no common factor greater than 1.

HCF(a, b, c) = 1

Examples:

(3, 4, 5)
(5, 12, 13)
(8, 15, 17)

Non-primitive triple

A triple is non-primitive if all three numbers have a common factor greater than 1.

HCF(a, b, c) > 1

Examples:

(6, 8, 10), HCF = 2
(9, 12, 15), HCF = 3
(12, 16, 20), HCF = 4

How to check whether a triple is primitive

First check that a² + b² = c².

Find the HCF of all three numbers.

If HCF = 1, the triple is primitive. If HCF > 1, it is non-primitive.

Can a non-primitive triple be reduced to a primitive triple?

Yes. Divide all three numbers by their greatest common divisor.

(9, 12, 15) ÷ 3 = (3, 4, 5)

The resulting triple is primitive.

Generating More Primitive Triples

One formula for generating Pythagorean triples uses two positive integers m and n, where m > n:

a = m² − n²
b = 2mn
c = m² + n²

To obtain a primitive triple, choose m and n that have no common factor greater than 1 and are not both odd.

Example

Let m = 2 and n = 1.

a = 2² − 1² = 3
b = 2 × 2 × 1 = 4
c = 2² + 1² = 5

Hence, the triple is (3, 4, 5).

This formula is a useful method for generating triples. The condition on m and n ensures the resulting triple is primitive.

Sum of the First n Odd Numbers

The sum of consecutive odd numbers follows a special pattern:

1 = 1²

1 + 3 = 2²

1 + 3 + 5 = 3²

1 + 3 + 5 + 7 = 4²

General formula

1 + 3 + 5 + ... + (2n − 1) = n²

The nth odd number is 2n − 1.

Using the sum of an arithmetic sequence:

S = n/2 × [1 + (2n − 1)]
S = n/2 × 2n
S = n²

Quick Revision

Concept	Remember
Pythagorean theorem	a² + b² = c²
Hypotenuse	Longest side, opposite the 90° angle
Base and perpendicular	Two sides forming the right angle
Baudhāyana triple	Three positive integers satisfying a² + b² = c²
Primitive triple	HCF of all three numbers is 1
Non-primitive triple	HCF of all three numbers is greater than 1
Scaled triple	Multiply every side by the same positive integer
Odd number sum	Sum of the first n odd numbers is n²

Common mistake: Do not assume that the base is always the shortest side. Only the hypotenuse is guaranteed to be the longest side.