Pythagorean Triples Formula
The Pythagorean triples formula, which consists of three numbers, is based on the famous right-angled theorem, also known as the Pythagorean theorem, a theorem proved by Pythagoras, a Greek mathematician. The list of these triples is frequently written in the form of three values or measurements, a, b and c, and they are always expressed in units of length.
The Pythagorean theorem states that,
\( c^2~=~a^2~+~b^2 \)
Where,
a – base of a right-angled triangle
b – perpendicular of the right angled triangle
c – hypotenuse of a right-angled triangle
Rapid Recall
Pythagorean Triples Formula: \( c^2~=~a^2~+~b^2 \)
Solved Examples
Example 1:
Prove whether (13, 84, 85) is a Pythagorean triple.
Solution:
The given triple = (13, 84, 85)
So, a = 13, b = 84 and c = 85
Applying the formula for Pythagorean triples, \( c^2~=~a^2~+~b^2 \)
Left hand side calculation: \( c^2~=~85^2~=~7225 \)
Right hand side calculation: \( a^2~+~b^2~=~13^2~+~84^2~=~169~+~7056~=~7225 \)
Left hand side calculation = Right hand side calculation
Therefore, (13, 84, 85) is a Pythagorean triple.
Example 2:
Check if (3, 4, 5) is a Pythagorean triple.
Solution:
The given triple = (3, 4, 5)
So that indicates, a = 3, b = 4 and c = 5
Applying the Pythagorean triples formula, \( c^2~=~a^2~+~b^2 \)
Left hand side calculation: \( c^2~=~5^2~=~25 \)
Right hand side calculation: \( a^2~+~b^2~=~3^2~+~4^2~=~9~+~16~=~25 \)
Left hand side calculation = Right hand side calculation
Hence, (3, 4, 5) is a Pythagorean triple.
Example 3:
Determine the value of ‘x’ using the Pythagorean triple formula if (x, 56, 65) is a Pythagorean triple.
Solution:
By using the formula of Pythagorean triples:
\( c^2~=~a^2~+~b^2 \)
Substituting ‘a’ by ‘x’, ‘b’ by ‘56’, and ‘c’ by ‘65’ in the formula we have,
\( 65^2~=~x^2~+~56^2 \) (Substitute the values of a, b and c)
\( 4225~=~x^2~+~3136 \)
\( x^2~=~4225~-~3136 \) (Subtract)
\( x^2~=~1089 \)
\( x~=~\sqrt{1089} \) (Square root of 1089 is 33)
\( x~=~33 \) (Simplify)
Hence, the value of ‘x’ is 33.
The numbers in a Pythagorean triples represent the sides of a right triangle, and the length of sides cannot have negative values. So, the numbers in a Pythagorean triples do not have a negative value.
The Pythagorean triples are linked to right angle triangles.
The most common appearing Pythagorean triples are:
(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97).