Introduction

A Pythagorean triplet is a set of three positive integers (a, b, c) that satisfy the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In mathematical terms, it can be expressed as a² + b² = c².

Pythagorean Triplet with a = 6

To find a Pythagorean triplet with one member as 6, we can start by assuming a = 6 and then try to find the other two numbers, b and c, that satisfy the Pythagorean theorem.

We can substitute the value of a in the equation a² + b² = c²:
6² + b² = c²
36 + b² = c²

Pythagorean Triplet with a = 6 and c = 14

Now, we need to find the value of b that satisfies the equation. Let's assume c = 14 and solve for b:

36 + b² = 14²
36 + b² = 196
b² = 196 - 36
b² = 160
b = √160
b ≈ 12.65

So, when a = 6 and c = 14, the value of b is approximately 12.65.

Checking if it forms a Pythagorean Triplet

To confirm if these values of a, b, and c form a Pythagorean triplet, we can substitute them back into the Pythagorean theorem equation:

6² + 12.65² = 14²
36 + 159.9225 = 196
195.9225 ≈ 196

The values of a = 6, b ≈ 12.65, and c = 14 approximately satisfy the Pythagorean theorem, confirming that they form a Pythagorean triplet.

Conclusion

In conclusion, a Pythagorean triplet with one member as 6 can be found with the values of a = 6, b ≈ 12.65, and c = 14. These values satisfy the Pythagorean theorem equation and form a right-angled triangle.

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