HOW TO FIND PYTHAGOREAN TRIPLET FROM ONE NUMBER
Consider the following,
32 + 42 = 52
The collection of numbers 3, 4 and 5 is known as Pythagorean triplet.
Relationship between Pythagorean Triplet :
Square of larger number
= Sum of squares of other two small numbers
If the given numbers will have the above relationship, we can say the given numbers are pythagorean triplets.
For any natural number m > 1, we have
(2m)2 + (m2 - 1)2 + (m2 + 1)2
So, 2m, (m2 - 1) and (m2 + 1) forms a Pythagorean triplet.
Example 1 :
Find the Pythagorean triplet in which one number is 8.
Solution :
We can get the Pythagorean triplet by using the general form 2m, (m2 - 1), (m2 + 1).
Let us consider the given number as 2m
2m = 8
m = 4
|
(m2 - 1) = (42 - 1) = 16 - 1 = 15 |
(m2 + 1) = (42 + 1) = 16 + 1 = 17 |
The triplet is 8, 15 and 17.
Verifying the relationship :
172 = 152 + 82
289 = 225 + 64
289 = 289
Example 2 :
Find the Pythagorean triplet in which one number is 12.
Solution :
We can get the Pythagorean triplet by using the general form 2m, (m2 - 1), (m2 + 1).
Let us consider the given number as 2m
2m = 12
m = 6
|
(m2 - 1) = (62 - 1) = 36 - 1 = 35 |
(m2 + 1) = (62 + 1) = 36 + 1 = 37 |
The triplet is 12, 35 and 37.
Verifying the relationship :
372 = 352 + 122
1369 = 1225 + 144
1369 = 1369
Example 3 :
Find the Pythagorean triplet in which one number is 14.
Solution :
We can get the Pythagorean triplet by using the general form 2m, (m2 - 1), (m2 + 1).
Let us consider the given number as 2m
2m = 14
m = 7
|
(m2 - 1) = (72 - 1) = 49 - 1 = 48 |
(m2 + 1) = (72 + 1) = 49 + 1 = 50 |
The triplet is 14, 48 and 50.
Verifying the relationship :
502 = 482 + 142
2500 = 2304 + 196
2500 = 2500
Example 4 :
Find the Pythagorean triplet in which one number is 6.
Solution :
We can get the Pythagorean triplet by using the general form 2m, (m2 - 1), (m2 + 1).
Let us consider the given number as 2m
2m = 6
m = 3
|
(m2 - 1) = (32 - 1) = 9 - 1 = 8 |
(m2 + 1) = (32 + 1) = 9 + 1 = 10 |
The triplet is 6, 8 and 10.
Verifying the relationship :
102 = 82 + 62
100 = 64 + 36
100 = 100
Example 5 :
Find the Pythagorean triplet in which one number is 16.
Solution :
We can get the Pythagorean triplet by using the general form 2m, (m2 - 1), (m2 + 1).
Let us consider the given number as 2m
2m = 16
m = 8
|
(m2 - 1) = (82 - 1) = 64 - 1 = 63 |
(m2 + 1) = (82 + 1) = 64 + 1 = 65 |
The triplet is 16, 63 and 65.
Verifying the relationship :
652 = 632 + 162
4225 = 3969 + 256
4225 = 4225
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