Pythagorean Triplets – Complete Guide for Competitive Exams
Pythagorean Triplets are very important in geometry and are frequently used in SSC, RRB NTPC, Banking, and other aptitude exams. A set of three numbers is called a Pythagorean Triplet when it satisfies the Pythagoras theorem.
Here, c represents the hypotenuse of a right-angled triangle and a and b are the other two sides.
Example
Consider the triplet (3,4,5)
- 3² = 9
- 4² = 16
- 5² = 25
- Since 9 + 16 = 25 → It satisfies the formula.
Most Important Pythagorean Triplets
| Type | a | b | c | Verification |
|---|---|---|---|---|
| Primitive | 3 | 4 | 5 | 9 + 16 = 25 |
| Primitive | 5 | 12 | 13 | 25 + 144 = 169 |
| Primitive | 8 | 15 | 17 | 64 + 225 = 289 |
| Derived | 6 | 8 | 10 | 36 + 64 = 100 |
Shortcut Method to Generate Triplets
Using Euclid's Formula we can generate infinite Pythagorean triplets. Choose two integers m and n such that m > n.
b = 2mn
c = m² + n²
Example:
- Let m = 2, n = 1
- a = 2² − 1² = 3
- b = 2×2×1 = 4
- c = 2² + 1² = 5
Hence the triplet is (3,4,5).
Important Properties
2️⃣ In every primitive triplet one number is always even.
3️⃣ The largest number in a triplet is always the hypotenuse.
Quick Triplets for Exams
- (3,4,5)
- (5,12,13)
- (7,24,25)
- (8,15,17)
- (9,40,41)
Memorizing these triplets helps solve geometry problems much faster in competitive exams.
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