STATEMENT-1 : The triangle with vertices (1, 3, 5), (2, 4, 6) and (0, 5, 7) must be a right angle triangle.
and
STATEMENT-2 : If the dot product of two non-zero vectors is zero then they must be perpendicular.

Options

A. Statement-1 is True, Statement-2 is true, Statement- is a correct explanation for Statement -1
B. Statement-1 is True, Statement-2 is true, Statement- is NOT a correct explanation for Statement -1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is true

Answer

To determine whether the triangle with vertices \( A(1, 3, 5) \), \( B(2, 4, 6) \), and \( C(0, 5, 7) \) is a right triangle, we will first find the vectors representing the sides of the triangle and then check if any two sides are perpendicular using the dot product.

### Step 1: Define the vertices
Let:
- \( A = (1, 3, 5) \)
- \( B = (2, 4, 6) \)
- \( C = (0, 5, 7) \)

### Step 2: Find the vectors representing the sides of the triangle
We can find the vectors \( \vec{AB} \), \( \vec{BC} \), and \( \vec{AC} \).

1. **Vector \( \vec{AB} \)**:
\[ \vec{AB} = B - A = (2 - 1, 4 - 3, 6 - 5) = (1, 1, 1) \]

2. **Vector \( \vec{BC} \)**:
\[ \vec{BC} = C - B = (0 - 2, 5 - 4, 7 - 6) = (-2, 1, 1) \]

3. **Vector \( \vec{AC} \)**:
\[ \vec{AC} = C - A = (0 - 1, 5 - 3, 7 - 5) = (-1, 2, 2) \]

### Step 3: Check for perpendicularity using the dot product
To check if any two vectors are perpendicular, we will calculate the dot product of \( \vec{AB} \) and \( \vec{BC} \).

\[ \vec{AB} \cdot \vec{BC} = (1, 1, 1) \cdot (-2, 1, 1) = 1 \cdot (-2) + 1 \cdot 1 + 1 \cdot 1 = -2 + 1 + 1 = 0 \]

Since the dot product \( \vec{AB} \cdot \vec{BC} = 0 \), the vectors \( \vec{AB} \) and \( \vec{BC} \) are perpendicular.

### Step 4: Conclusion
Since we have found that two sides of the triangle are perpendicular, triangle \( ABC \) is a right triangle.

### Statement Evaluation
- **Statement 1**: True, as triangle \( ABC \) is a right triangle.
- **Statement 2**: True, as the dot product of two non-zero vectors being zero indicates that they are perpendicular.

Thus, both statements are true, and Statement 2 correctly explains Statement 1.

### Final Answer
Both statements are true, and Statement 2 is the correct explanation for Statement 1.

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