Pythagorean Triplets:
Pythagorean triplets are sets of three positive integers that satisfy the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

The Pythagorean Theorem:
The Pythagorean theorem can be written as:
a² + b² = c²
Where a and b are the lengths of the two shorter sides (also known as the legs) of the right-angled triangle, and c is the length of the longest side (also known as the hypotenuse).

One Member is 10:
Let's find the Pythagorean triplet where one member is 10. We need to find two other positive integers that, when squared and summed, will equal 10 squared (100).

We can start by assuming one of the other members, let's say 'a', and write the equation:
a² + 10² = c²

Calculating 'c':
By rearranging the equation, we can solve for 'c':
c² = a² + 100
c = √(a² + 100)

Calculating 'a' and 'c':
To find the possible values of 'a' and 'c', we can substitute different values for 'a' and calculate 'c'. We will look for integers that satisfy the equation.

Example:
Let's assume 'a' as 6:
c = √(6² + 100) = √(36 + 100) = √136 ≈ 11.66

Here, 'c' is not an integer, so 'a' = 6 does not satisfy the Pythagorean theorem.

Calculating 'a' and 'c' continued:
We can continue this process by trying different values of 'a' until we find a Pythagorean triplet where one member is 10.

Example:
Let's assume 'a' as 8:
c = √(8² + 100) = √(64 + 100) = √164 ≈ 12.81

Again, 'c' is not an integer, so 'a' = 8 does not satisfy the Pythagorean theorem.

Final Calculation:
By trying different values for 'a', we find that there is no Pythagorean triplet where one member is exactly 10. The closest we can get is when 'a' = 6, where the values of 'a', 'b', and 'c' are approximately 6, 10, and 11.66, respectively.

Therefore, there is no Pythagorean triplet with one member as exactly 10.