If `a+b+c=2s, " then " [(s - a)^2 + (s -b)^2 + (s - c)^2 + s^2] =?`

Options

A. `(s^2 - a^2 – b^2-c^2)`
B. `(s^2 + a^2 + b^2+c^2)`
C. `(s^2 + b^2+c^2)`
D. `(4s^2 - a^2 – b^2-c^2)`

Answer

To solve the equation given that \( a + b + c = 2s \), we need to find the value of the expression:

\[ (s - a)^2 + (s - b)^2 + (s - c)^2 + s^2 \]

### Step-by-Step Solution:

1. **Expand Each Square**:
We start by expanding each term in the expression:
\[ (s - a)^2 = s^2 - 2sa + a^2 \]
\[ (s - b)^2 = s^2 - 2sb + b^2 \]
\[ (s - c)^2 = s^2 - 2sc + c^2 \]

2. **Combine the Expanded Terms**:
Now, we can substitute these expansions back into the original expression:
\[ (s - a)^2 + (s - b)^2 + (s - c)^2 + s^2 = (s^2 - 2sa + a^2) + (s^2 - 2sb + b^2) + (s^2 - 2sc + c^2) + s^2 \]

3. **Simplify the Expression**:
Combine like terms:
\[ = s^2 + s^2 + s^2 + s^2 - 2sa - 2sb - 2sc + a^2 + b^2 + c^2 \]
\[ = 4s^2 - 2s(a + b + c) + (a^2 + b^2 + c^2) \]

4. **Substitute \( a + b + c \)**:
We know from the problem statement that \( a + b + c = 2s \). Substitute this into the expression:
\[ = 4s^2 - 2s(2s) + (a^2 + b^2 + c^2) \]
\[ = 4s^2 - 4s^2 + (a^2 + b^2 + c^2) \]

5. **Final Result**:
The terms \( 4s^2 - 4s^2 \) cancel out, leaving us with:
\[ = a^2 + b^2 + c^2 \]

Thus, the value of the expression \( (s - a)^2 + (s - b)^2 + (s - c)^2 + s^2 \) is:

\[ \boxed{a^2 + b^2 + c^2} \]