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The formula for the Pythagorean triples consists of three numbers that satisfy the conditions of the Pythagorean theorem. These triples are collectively known as the Pythagorean triples, which are commonly written as (a, b, c). A triangle with sides that have these triples as their dimensions is said to be a Pythagorean triangle....Read MoreRead Less

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

Pythagorean Triples Formula: \( c^2~=~a^2~+~b^2 \)

**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.

Frequently Asked Questions

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).