True (T) or false (F)
If one member of a pythagorean triplet is 2m, then the other two members are

Options

A. `m , m^2+1`
B. `m^2 + 1,m^2-1`
C. `m^2,m^2-1`
D. `m^2,m+1`

Answer

To determine whether the statement is true or false, we need to analyze the concept of Pythagorean triplets. A Pythagorean triplet consists of three positive integers \(a\), \(b\), and \(c\) such that they satisfy the equation:

\[ a^2 + b^2 = c^2 \]

where \(c\) is the hypotenuse (the longest side) of a right triangle, and \(a\) and \(b\) are the other two sides.

### Step 1: Identify the given member
We are given that one member of the Pythagorean triplet is \(2m\).

### Step 2: Set up the Pythagorean theorem
Let’s denote the other two members of the triplet as \(a\) and \(b\). According to the Pythagorean theorem, we can express this as:

\[ (2m)^2 + a^2 = b^2 \]

### Step 3: Substitute and simplify
Substituting \(2m\) into the equation gives us:

\[ 4m^2 + a^2 = b^2 \]

### Step 4: Rearranging the equation
We can rearrange this equation to find a relationship between \(a\) and \(b\):

\[ b^2 - a^2 = 4m^2 \]

### Step 5: Factor the difference of squares
The left side can be factored using the difference of squares:

\[ (b - a)(b + a) = 4m^2 \]

### Step 6: Determine possible integer values
To satisfy this equation, \(b - a\) and \(b + a\) must be factors of \(4m^2\).

### Step 7: Find specific values
Assuming \(b - a = 2m\) and \(b + a = 2m\), we can solve these equations:

1. \(b - a = 2m\)
2. \(b + a = 2m\)

Adding these two equations:

\[ 2b = 4m \implies b = 2m \]

Subtracting the first from the second:

\[ 2a = 0 \implies a = 0 \]

This means that \(b\) cannot be equal to \(2m\) if we are looking for positive integers.

### Conclusion
Since we cannot find two positive integers \(a\) and \(b\) such that one of the triplet members is \(2m\) while satisfying the Pythagorean theorem, the statement is **False (F)**.