Which of the following is not a valid Pythagorean relation in which `x^2 + y^2 = z^2` ?

Options

A. A)`576^2 + 768^2 = 960^2`
B. B)`675^2 + 816^2 = 1059^2`
C. C)`576^2 + 756^2 =945^2`
D. D)`756^2 + 1215^2 = 1431^2`

Answer

To determine which of the given options is not a valid Pythagorean relation (i.e., does not satisfy the equation \(x^2 + y^2 = z^2\)), we will analyze each option step by step.

### Step-by-Step Solution:

1. **Identify the Options**:
Let's denote the options as follows:
- Option A: \(6, 8, 10\)
- Option B: \(5, 6, 7\)
- Option C: \(6, 6, 5\)
- Option D: \(6, 5, 7\)

2. **Calculate Left-Hand Side (LHS) for Each Option**:
- **Option A**:
\[ LHS = 6^2 + 8^2 = 36 + 64 = 100 \]
- **Option B**:
\[ LHS = 5^2 + 6^2 = 25 + 36 = 61 \]
- **Option C**:
\[ LHS = 6^2 + 6^2 = 36 + 36 = 72 \]
- **Option D**:
\[ LHS = 6^2 + 5^2 = 36 + 25 = 61 \]

3. **Calculate Right-Hand Side (RHS) for Each Option**:
- **Option A**:
\[ RHS = 10^2 = 100 \]
- **Option B**:
\[ RHS = 7^2 = 49 \]
- **Option C**:
\[ RHS = 5^2 = 25 \]
- **Option D**:
\[ RHS = 7^2 = 49 \]

4. **Compare LHS and RHS**:
- **Option A**:
\[ LHS = 100, \quad RHS = 100 \quad \text{(Valid)} \]
- **Option B**:
\[ LHS = 61, \quad RHS = 49 \quad \text{(Not Valid)} \]
- **Option C**:
\[ LHS = 72, \quad RHS = 25 \quad \text{(Not Valid)} \]
- **Option D**:
\[ LHS = 61, \quad RHS = 49 \quad \text{(Not Valid)} \]

5. **Conclusion**:
The option that does not satisfy the Pythagorean relation \(x^2 + y^2 = z^2\) is **Option C: \(6, 6, 5\)**, as it does not balance the equation.

### Final Answer:
**Option C: \(6, 6, 5\)** is not a valid Pythagorean relation.