If ` a + b + c = 2s` then
`((s - a)^(2) + (s - b)^(2) +(s - c)^(2) +s^(2))/(a^(2) + b^(2) + c^(2))` is equal to

Options

A. A)`a^(2) + b^(2) + c^(2)`
B. B)0
C. C)1
D. D)2

Answer

To solve the problem step by step, we start with the given equation and proceed through the calculations systematically.

### Step 1: Understand the Given Equation
We are given that \( a + b + c = 2s \).

### Step 2: Substitute Values for a, b, and c
To simplify our calculations, we can choose specific values for \( a \), \( b \), and \( c \). Let's set \( a = 2 \), \( b = 2 \), and \( c = 2 \).

### Step 3: Calculate s
Using the equation \( a + b + c = 2s \):
\[ 2 + 2 + 2 = 2s \implies 6 = 2s \implies s = 3 \]

### Step 4: Substitute s, a, b, and c into the Expression
We need to evaluate the expression:
\[ \frac{(s - a)^2 + (s - b)^2 + (s - c)^2 + s^2}{a^2 + b^2 + c^2} \]
Substituting the values \( s = 3 \), \( a = 2 \), \( b = 2 \), and \( c = 2 \):
\[ \frac{(3 - 2)^2 + (3 - 2)^2 + (3 - 2)^2 + 3^2}{2^2 + 2^2 + 2^2} \]

### Step 5: Calculate Each Part of the Numerator
Calculating \( (3 - 2)^2 \):
\[ (3 - 2)^2 = 1^2 = 1 \]
Thus, we have:
\[ 1 + 1 + 1 + 3^2 = 1 + 1 + 1 + 9 = 12 \]

### Step 6: Calculate the Denominator
Calculating \( a^2 + b^2 + c^2 \):
\[ 2^2 + 2^2 + 2^2 = 4 + 4 + 4 = 12 \]

### Step 7: Combine the Results
Now we can combine the results:
\[ \frac{12}{12} = 1 \]

### Final Answer
Thus, the expression evaluates to:
\[ \boxed{1} \]