STATEMENT-1 : The points (2, 3, 5), (7, 5, 7) and (-3, 1, 3) are vertices of an equilateral triangle.
and
STATEMENT-2 : The triangle with equal sides is called an equilateral triangle.

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 points (2, 3, 5), (7, 5, 7), and (-3, 1, 3) form an equilateral triangle, we will calculate the lengths of the sides of the triangle formed by these points using the distance formula in three-dimensional geometry.

### Step 1: Identify the points
Let:
- Point A = (2, 3, 5)
- Point B = (7, 5, 7)
- Point C = (-3, 1, 3)

### Step 2: Use the distance formula
The distance \(d\) between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in three-dimensional space is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

### Step 3: Calculate the lengths of the sides

#### Length AB
Using the points A and B:
\[ AB = \sqrt{(7 - 2)^2 + (5 - 3)^2 + (7 - 5)^2} \]
\[ = \sqrt{5^2 + 2^2 + 2^2} = \sqrt{25 + 4 + 4} = \sqrt{33} \]

#### Length BC
Using the points B and C:
\[ BC = \sqrt{(7 - (-3))^2 + (5 - 1)^2 + (7 - 3)^2} \]
\[ = \sqrt{(7 + 3)^2 + (5 - 1)^2 + (7 - 3)^2} = \sqrt{10^2 + 4^2 + 4^2} \]
\[ = \sqrt{100 + 16 + 16} = \sqrt{132} \]

#### Length AC
Using the points A and C:
\[ AC = \sqrt{(-3 - 2)^2 + (1 - 3)^2 + (3 - 5)^2} \]
\[ = \sqrt{(-5)^2 + (-2)^2 + (-2)^2} = \sqrt{25 + 4 + 4} = \sqrt{33} \]

### Step 4: Compare the lengths
We have:
- \(AB = \sqrt{33}\)
- \(BC = \sqrt{132}\)
- \(AC = \sqrt{33}\)

From the calculations, we find that:
- \(AB = AC\)
- \(BC \neq AB\) and \(BC \neq AC\)

### Conclusion
Since not all sides are equal, the triangle formed by the points (2, 3, 5), (7, 5, 7), and (-3, 1, 3) is not an equilateral triangle. Thus, **Statement 1 is false**.

**Statement 2 is true** because a triangle with equal sides is indeed called an equilateral triangle.

### Final Answer
- Statement 1: False
- Statement 2: True

### Options
The correct option is **Option 4**: Statement 1 is false, Statement 2 is true.

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