STATEMENT-1 The points A(2, 9, 12), B(1, 8, 8), C(-2, 11, 8) and D(-1, 12, 12) are the vertices of rhombus.
and
STATEMENT-2 : `AB = BC = CD = DA and AC ne BD` then the quadrilateral ABCD is called a rhombus.

Options

A. (a)Statement-1 is True, Statement-2 is true, (a)Statement- is a correct explanation for Statement -1
B. (b)Statement-1 is True, Statement-2 is true, Statement- is NOT a correct explanation for Statement -1
C. (c)Statement-1 is True, Statement-2 is False
D. (d)Statement-1 is False, Statement-2 is true

Answer

To determine whether the points A(2, 9, 12), B(1, 8, 8), C(-2, 11, 8), and D(-1, 12, 12) form a rhombus, we need to verify the following conditions:

1. All sides of the quadrilateral must be equal (AB = BC = CD = DA).
2. The diagonals must not be equal (AC ≠ BD).

### Step 1: Calculate the lengths of the sides

We will use the distance formula in three dimensions, which is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

#### Length of AB
Coordinates of A = (2, 9, 12) and B = (1, 8, 8)

\[ AB = \sqrt{(1 - 2)^2 + (8 - 9)^2 + (8 - 12)^2} \]
\[ = \sqrt{(-1)^2 + (-1)^2 + (-4)^2} \]
\[ = \sqrt{1 + 1 + 16} = \sqrt{18} \]

#### Length of BC
Coordinates of B = (1, 8, 8) and C = (-2, 11, 8)

\[ BC = \sqrt{(-2 - 1)^2 + (11 - 8)^2 + (8 - 8)^2} \]
\[ = \sqrt{(-3)^2 + (3)^2 + 0^2} \]
\[ = \sqrt{9 + 9 + 0} = \sqrt{18} \]

#### Length of CD
Coordinates of C = (-2, 11, 8) and D = (-1, 12, 12)

\[ CD = \sqrt{(-1 + 2)^2 + (12 - 11)^2 + (12 - 8)^2} \]
\[ = \sqrt{(1)^2 + (1)^2 + (4)^2} \]
\[ = \sqrt{1 + 1 + 16} = \sqrt{18} \]

#### Length of DA
Coordinates of D = (-1, 12, 12) and A = (2, 9, 12)

\[ DA = \sqrt{(2 + 1)^2 + (9 - 12)^2 + (12 - 12)^2} \]
\[ = \sqrt{(3)^2 + (-3)^2 + 0^2} \]
\[ = \sqrt{9 + 9 + 0} = \sqrt{18} \]

### Step 2: Check the lengths of the sides
From the calculations, we have:
- AB = √18
- BC = √18
- CD = √18
- DA = √18

Thus, all sides are equal: \(AB = BC = CD = DA = \sqrt{18}\).

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

#### Length of AC
Coordinates of A = (2, 9, 12) and C = (-2, 11, 8)

\[ AC = \sqrt{(-2 - 2)^2 + (11 - 9)^2 + (8 - 12)^2} \]
\[ = \sqrt{(-4)^2 + (2)^2 + (-4)^2} \]
\[ = \sqrt{16 + 4 + 16} = \sqrt{36} = 6 \]

#### Length of BD
Coordinates of B = (1, 8, 8) and D = (-1, 12, 12)

\[ BD = \sqrt{(-1 - 1)^2 + (12 - 8)^2 + (12 - 8)^2} \]
\[ = \sqrt{(-2)^2 + (4)^2 + (4)^2} \]
\[ = \sqrt{4 + 16 + 16} = \sqrt{36} = 6 \]

### Step 4: Check the lengths of the diagonals
From the calculations, we have:
- AC = 6
- BD = 6

Since \(AC = BD\), the diagonals are equal.

### Conclusion
Since all sides are equal but the diagonals are equal as well, the quadrilateral ABCD does not satisfy the condition for being a rhombus (where the diagonals must not be equal). Therefore, Statement 1 is false.

### Summary
- Statement 1: False
- Statement 2: True (as it correctly describes the properties of a rhombus)