Know and learn complete details about Pythagorean triplets. Problems based on Pythagoras theorem can be solved easily if you know Pythagorean triplets.
Every time instead of applying Pythagoras theorem for finding sides of right angled triangle you can make use of this triplets.
First, it is required to know what is Pythagoras theorem. After that, we will study its triplets. Pythagoras theorem definition is given as “For a given any right angled triangle, hypotenuse square will be always equal to the sum of squares of remaining two sides”.
Here, Hypotenuse = RQ and the remaining two sides are PQ and PR.
Therefore, according to Pythagoras theorem statement, we can write it as
{ hypotenuse }^{ 2 } = { side }^{ 2 } + { side }^{ 2 }
{ RQ }^{ 2 } = { PQ }^{ 2 } + { PR }^{ 2 }
Now we will study about Pythagorean triplets:
The numbers which satisfies Pythagoras theorem are called Pythagorean triplets.
Example:
3, 4, and 5 is one of the important Pythagorean triplets because { 5 }^{ 2 } = { 3 }^{ 2 } + { 4 }^{ 2 } , here the greatest number will be always hypotenuse.
The Formula for finding Pythagorean triplets:
The general notation of Pythagorean triplets is { P }^{ 2 } – 1, 2P, { P }^{ 2 } + 1 where P is an integer. 2P represents the least length (side) and { P }^{ 2 } + 1 represents hypotenuse { P }^{ 2 } – 1 represents another side of right angled triangle.
Example 1:
If 2P = 8
P = 4
Then { P }^{ 2 } + 1 = { 4 }^{ 2 } + 1 = 17
{ P }^{ 2 } – 1 = { 4 }^{ 2 } -1 = 15
Therefore, the triplets are 15, 8, 17 because { 17 }^{ 2 } = { 8 }^{ 2 } + { 15 }^{ 2 }
Example 2:
If 2P = 14
P = 7
Then { P }^{ 2 } + 1 = { 7 }^{ 2 } + 1 = 50
{ P }^{ 2 } – 1 = { 7 }^{ 2 } -1 = 48
Therefore, the triplets are 48, 14, 50 because { 50 }^{ 2 } = { 48 }^{ 2 } + { 14 }^{ 2 }
Some important Pythagorean triplets:
3, 4, 5
8, 6, 10
15, 8, 17
24, 10, 26
35, 12, 37
48, 14, 50
63, 16, 65
80, 18, 82
99, 20, 101
120, 22, 122
143, 24, 145
168, 26, 170
195, 28, 197
224, 30, 226
255, 32, 257
288, 34, 290
323, 36, 325
360, 38, 362
399, 40, 401
440, 42, 442
483, 44, 485
528, 46, 530
575, 48, 577
624, 50, 626
675, 52, 677
728, 54, 730
783, 56, 785
840, 58, 842
899, 60, 901
960, 62, 962
1023, 64, 1025
1088, 66, 1090
1155, 68, 1157
1224, 70, 1226
1295, 72, 1297
1368, 74, 1370
1443, 76, 1445
1520, 78, 1522
1599, 80, 1601
1680, 82, 1682
1763, 84, 1765
1848, 86, 1850
1935, 88, 1937
2024, 90, 2026
2115, 92, 2117
2208, 94, 2210
2303, 96, 2305
2400, 98, 2402
2499, 100, 2501
Note:
Try to remember at least first 5 to 6 triplets
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