Statement I: In any right angled triangle `(a^(2)+b^(2)+c^(2))/(R^(2))` is always equal to 8.
Statement II: `a ^(2)=b^(2) +c^(2)`

Options

A. a) Both Statement I and Statement II are correct and Statement II is the correct explanation of Statement I
B. b) Both Statement I and Statement II are correct and Statement II is not the correct explanation of Statement I
C. c) Statement I is correct but Statement II is incorrect
D. d) Statement I is incorrect but Statement II is correct

Answer

To solve the problem, we will analyze the two statements regarding a right-angled triangle.

### Step 1: Understand the Right-Angled Triangle
Let triangle ABC be a right-angled triangle with angle A = 90 degrees. In a right-angled triangle, the Pythagorean theorem holds true, which states:
\[ a^2 = b^2 + c^2 \]
where:
- \( a \) is the length of the hypotenuse,
- \( b \) and \( c \) are the lengths of the other two sides.

### Step 2: Analyze Statement II
From the Pythagorean theorem, we can confirm that:
\[ a^2 = b^2 + c^2 \]
This means Statement II is correct.

### Step 3: Calculate \( \frac{a^2 + b^2 + c^2}{R^2} \)
To analyze Statement I, we need to calculate \( \frac{a^2 + b^2 + c^2}{R^2} \).

Using the Pythagorean theorem, we can express \( a^2 + b^2 + c^2 \):
\[ a^2 + b^2 + c^2 = a^2 + (a^2 - c^2) + c^2 = 2a^2 \]
Thus, we have:
\[ \frac{a^2 + b^2 + c^2}{R^2} = \frac{2a^2}{R^2} \]

### Step 4: Relate \( a \) to the Circumradius \( R \)
In a right-angled triangle, the circumradius \( R \) is given by:
\[ R = \frac{a}{2} \]
Substituting this into our equation:
\[ R^2 = \left(\frac{a}{2}\right)^2 = \frac{a^2}{4} \]

### Step 5: Substitute \( R^2 \) into the Equation
Now substituting \( R^2 \) back into our equation:
\[ \frac{2a^2}{R^2} = \frac{2a^2}{\frac{a^2}{4}} = \frac{2a^2 \cdot 4}{a^2} = 8 \]

### Conclusion
Thus, we find that:
\[ \frac{a^2 + b^2 + c^2}{R^2} = 8 \]
This confirms that Statement I is also correct.

### Final Result
Both statements are correct, and Statement II is the correct explanation for Statement I. Therefore, the correct option is:
**Option A: Both Statement I and Statement II are correct, and Statement II is the correct explanation of Statement I.**

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