What is a Pythagorean triple?

Pythagorean triple (PT) can be defined as a collection of three positive whole numbers that perfectly satisfy the Pythagorean theorem: a2 + b2 = c2.

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This set of numbers room usually the three side lengths of a right triangle. Pythagorean triples are represented as: (a, b, c), where, a = one leg; b = an additional leg; and also c = hypotenuse.

There space two types of Pythagorean triples:

Primitive Pythagorean triplesNon-primitive Pythagorean triples

Primitive Pythagorean triples

A primitive Pythagorean triple is a reduced set of the positive values that a, b, and c with a common factor other than 1. This type of triple is always composed that one also number and also two strange numbers.

For example, (3, 4, 5) and also (5, 12, 13) are instances of primitive Pythagorean triples because each collection has a common factor that 1 and additionally satisfies the

Pythagorean theorem: a2 + b2 = c2.

(3, 4, 5) → GCF =1

a2 + b2 = c2

32 + 42 = 52

9 + 16 = 25

25 = 25

(5, 12, 13) → GCF = 1

a2 + b2 = c2

52 + 122 = 132

25 + 144 = 169

169 = 169

Non-primitive Pythagorean triples

A non-primitive Pythagorean triple, also known as the imperative Pythagorean triple, is a collection of confident values the a, b, and also c with a common factor greater than 1. In other words, the three sets of optimistic values in a non-primitive Pythagorean triple space all even numbers.

Examples that non-primitive Pythagorean triples include: (6,8,10), (32,60,68), (16, 30, 34) etc.

(6,8,10) → GCF the 6, 8 and also 10 = 2.

a2 + b2 = c2

62 + 82 = 102

36 + 64 = 100

= 100(32,60,68) → GCF that 32, 60 and also 68 = 4

a2 + b2 = c2

322 + 602 = 682

1,024 + 3,600 = 4,624

4,624 = 4,624

Other examples of commonly used Pythagorean triples include: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25) , (20, 21, 29) , (12, 35, 37), (9, 40, 41), (28, 45, 53), (11, 60, 61), (16, 63, 65), (33, 56, 65), (48, 55, 73), etc.

Properties the Pythagorean triples

From the over illustration of different types of Pythagorean triples, we make the complying with conclusions about Pythagorean triples:

A Pythagorean triple cannot have actually composed of just odd numbers.Similarly, a triple a Pythagorean triple deserve to never save on computer one strange number and two strange numbers.If (a, b, c) is a Pythagorean triple, then either a or b is the short or lengthy leg that the triangle, and c is the hypotenuse.

Pythagorean Triples Formula

The Pythagorean triples formula have the right to generate both primitive Pythagorean triples and non-primitive Pythagorean triples.

Pythagorean triples formula is offered as:

(a, b, c) = < (m2 − n2); (2mn); (m2 + n2)>

Where m and also n are two optimistic integers and also m > n

NOTE: If one member of the triple is known, us can attain the staying members by making use of the formula: (a, b, c) = < (m2-1), (2m), (m2+1)>.

Example 1

What is the Pythagorean triple the two positive numbers, 1 and 2?

Solution

Given the Pythagorean triples formula: (a, b, c) = (m2 − n2; 2mn; m2 + n2), where; m > n.

So, let m = 2 and n = 1.

Substitute the values of m and n into the formula.

⇒ a = 22 − 12 = 4 − 1 = 3

a =3

⇒ b = 2 × 2 × 1 = 4

b = 4

⇒ c = 22 + 12 = 4 + 1 = 5

c = 5

Apply the Pythagorean theorem to verify the (3,4,5) is indeed a Pythagorean triple

⇒ a2 + b2 = c2

⇒ 32 + 42 = 52

⇒ 9 + 16 = 25

⇒ 25 = 25.

Yes, that worked! Therefore, (3,4,5) is a Pythagorean triple.

Example 2

Generate a Pythagorean triple from two integers 5 and 3.

Solution

Since m need to be higher than n (m > n), let m= 5 and also n = 2.

a = m2 − n2

⇒a= (5)2 −(3)2 = 25−9

= 16

⇒ b = 2mn = 2 x 5 x 3

= 30

⇒ c = m2 + n2 = 32 + 52= 9 + 25= 34

Hence, (a, b, c) = (16, 30, 34).

Verify the answer.

⇒ a2 + b2 = c2

⇒ 162 + 302 = 342

⇒ 256 + 900 = 1,156

1,156 = 1,156 (True)

Therefore, (16, 30, 34) is undoubtedly a Pythagorean triple.

Example 3

Check if (17, 59, 65) is a Pythagorean triple.

Solution

Let, a = 17, b = 59, c = 65.

Test if, a2 + b2 = c2.

a2 + b2 ⇒ 172 + 592

⇒ 289 + 3481 = 3770

c2 = 652

= 4225

Since 3770 ≠ 4225, then (17, 59, 65) is not a Pythagorean triple.

Example 4

Find the possible value the ‘a’ in the following Pythagorean triple:(a, 35, 37).

Solution

Apply the Pythagorean equation a2 + b2 = c2.

a2 + 352 = 372.

a2 = 372−352=144. ​

√a2 = √144

a = 12.

Example 5

Find the Pythagorean triple the a best triangle whose hypotenuse is 17 cm.

Solution

(a, b, c) = < (m2-1), (2m), (m2+1)>

c = 17 = m2+1

17 – 1 = m2

m2 = 16

m = 4.

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Therefore,

b = 2m = 2 x 4

= 8

a = m2 – 1

= 42 – 1

= 15

Example 6

The smallest side of a ideal triangle is 20mm. Find the Pythagorean triple that the triangle.