Introduction:

In mathematics, a Pythagorean triplet is a set of three positive integers that satisfy the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, we are looking for a Pythagorean triplet for the number 11.

Pythagorean Theorem:

The Pythagorean theorem can be stated as follows:

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.

Formula:

The formula to calculate a Pythagorean triplet is:

a^2 + b^2 = c^2

Where 'a', 'b', and 'c' are the sides of the right-angled triangle.

Calculating the Pythagorean Triplet for 11:

To find a Pythagorean triplet for the number 11, we need to find three positive integers 'a', 'b', and 'c' that satisfy the equation a^2 + b^2 = c^2, where c is equal to 11.

Step 1: Start by assuming values for 'a' and 'b'. Let's assume 'a' to be the smallest possible positive integer, which is 1.

Step 2: Substitute the values of 'a' and 'c' into the equation and solve for 'b'.

1^2 + b^2 = 11^2
1 + b^2 = 121
b^2 = 120
b = √120

Step 3: Simplify the square root of 120.

b = √(2^3 * 3 * 5) = 2√30

Step 4: Verify if 'b' is a positive integer. In this case, 2√30 is not a positive integer.

Step 5: Repeat steps 1 to 4 by incrementing the value of 'a'. Let's assume 'a' to be 2.

Step 6: Substitute the values of 'a' and 'c' into the equation and solve for 'b'.

2^2 + b^2 = 11^2
4 + b^2 = 121
b^2 = 117
b = √117

Step 7: Simplify the square root of 117.

b = √(3^2 * 13) = 3√13

Step 8: Verify if 'b' is a positive integer. In this case, 3√13 is not a positive integer.

Step 9: Repeat steps 1 to 8 by incrementing the value of 'a'. Let's assume 'a' to be 3.

Step 10: Substitute the values of 'a' and 'c' into the equation and solve for 'b'.

3^2 + b^2 = 11^2
9 + b^2 = 121
b^2 = 112
b = √112

Step 11