Coordinates of the point O, P, Q and R are respectively (0,0,4),(4,6,2m),(2,0,2n) and (2,4,6). Let L,M,N and K be points on the sides OR, OP, PQ and QR respectively such that LMNK is a parallelogram whose two adjecent sides LM and side LK are each of length `sqrt2`. What are the values of m and n respectively ?

Options

A. 6,2
B. 1,3
C. 3,1
D. None of the above

Explanation

To solve the problem, we need to find the values of \( m \) and \( n \) given the coordinates of points \( O, P, Q, R \) and the conditions for the parallelogram \( LMNK \). Here are the steps to arrive at the solution:

### Step 1: Identify the Coordinates
The coordinates of the points are given as:
- \( O(0, 0, 4) \)
- \( P(4, 6, 2m) \)
- \( Q(2, 0, 2n) \)
- \( R(2, 4, 6) \)

### Step 2: Find the Midpoints
Let’s find the coordinates of points \( M \) and \( L \) using the midpoint formula. The midpoint \( M \) of segment \( OP \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]
Substituting \( O \) and \( P \):
\[ M = \left( \frac{0 + 4}{2}, \frac{0 + 6}{2}, \frac{4 + 2m}{2} \right) = \left( 2, 3, m + 2 \right) \]

The midpoint \( L \) of segment \( OR \) is:
\[ L = \left( \frac{0 + 2}{2}, \frac{0 + 4}{2}, \frac{4 + 6}{2} \right) = \left( 1, 2, 5 \right) \]

### Step 3: Calculate Lengths of Sides
We know that the lengths of adjacent sides \( LM \) and \( LK \) are each \( \sqrt{2} \).

**Length \( LM \)**:
Using the distance formula:
\[ LM = \sqrt{(x_M - x_L)^2 + (y_M - y_L)^2 + (z_M - z_L)^2} \]
Substituting the coordinates of \( L \) and \( M \):
\[ LM = \sqrt{(2 - 1)^2 + (3 - 2)^2 + ((m + 2) - 5)^2} = \sqrt{1^2 + 1^2 + (m - 3)^2} \]
Setting this equal to \( \sqrt{2} \):
\[ \sqrt{2 + (m - 3)^2} = \sqrt{2} \]
Squaring both sides:
\[ 2 + (m - 3)^2 = 2 \implies (m - 3)^2 = 0 \implies m = 3 \]

### Step 4: Find Length \( LK \)
Now we need to find \( LK \). The coordinates of \( K \) are on segment \( QR \):
\[ K = \left( \frac{2 + 2}{2}, \frac{0 + 4}{2}, \frac{2n + 6}{2} \right) = \left( 2, 2, n + 3 \right) \]

Now calculate \( LK \):
\[ LK = \sqrt{(2 - 1)^2 + (2 - 2)^2 + ((n + 3) - 5)^2} = \sqrt{1^2 + 0^2 + (n - 2)^2} \]
Setting this equal to \( \sqrt{2} \):
\[ \sqrt{1 + (n - 2)^2} = \sqrt{2} \]
Squaring both sides:
\[ 1 + (n - 2)^2 = 2 \implies (n - 2)^2 = 1 \implies n - 2 = \pm 1 \]
Thus, \( n = 3 \) or \( n = 1 \).

### Final Values
The values of \( m \) and \( n \) are:
\[ m = 3, \quad n = 3 \text{ or } n = 1 \]