## 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.You are watching: Which set represents a pythagorean triple**

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 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 =1a2 + b2 = c2

32 + 42 = 52

9 + 16 = 25

25 = 25

(5, 12, 13) → GCF = 1a2 + 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.

a2 + b2 = c2

62 + 82 = 102

36 + 64 = 100

= 100(32,60,68) → GCF that 32, 60 and also 68 = 4a2 + 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**:

## 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.