In a right angled triangle `a^(2)+b^(2)+c^(2)` is :

Options

A. `R^(2)`
B. `4R^(2)`
C. `8R^(2)`
D. none of these

Answer

To solve the problem, we need to find the value of \( a^2 + b^2 + c^2 \) in a right-angled triangle, where \( a \), \( b \), and \( c \) represent the lengths of the sides of the triangle.

### Step-by-step Solution:

1. **Understand the Triangle**: We have a right-angled triangle with sides \( a \), \( b \), and hypotenuse \( c \). According to the Pythagorean theorem, we know that:
\[ a^2 + b^2 = c^2 \]

2. **Identify the Circumradius**: In a right-angled triangle, the circumradius \( R \) is given by:
\[ R = \frac{c}{2} \]
Therefore, we can express \( c \) in terms of \( R \):
\[ c = 2R \]

3. **Substituting \( c \) in the Pythagorean Theorem**: We can substitute \( c \) in the Pythagorean theorem:
\[ a^2 + b^2 = (2R)^2 \]
This simplifies to:
\[ a^2 + b^2 = 4R^2 \]

4. **Calculate \( a^2 + b^2 + c^2 \)**: Now we need to calculate \( a^2 + b^2 + c^2 \):
\[ a^2 + b^2 + c^2 = a^2 + b^2 + (2R)^2 \]
Substituting \( a^2 + b^2 = 4R^2 \) into the equation:
\[ a^2 + b^2 + c^2 = 4R^2 + 4R^2 = 8R^2 \]

5. **Conclusion**: Therefore, the value of \( a^2 + b^2 + c^2 \) in a right-angled triangle is:
\[ a^2 + b^2 + c^2 = 8R^2 \]

### Final Answer:
The correct option is \( 8R^2 \).

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