The coordinates of the mid-points of the sides of `DeltaPQR`, are `(3a,0,0), (0,3b,0)` and `(0,0,3c)` respectively, then the area of `DeltaPQR` is

Options

A. `18sqrt(b^(2)c^(2)+c^(2)a^(2)+a^(2)b^(2))`
B. `9sqrt(b^(2)c^(2)+c^(2)a^(2)+a^(2)b^(2))`
C. `9/12sqrt(b^(2)c^(2)+c^(2)a^(2)+a^(2)b^(2))`
D. `9/2sqrt(b^(2)c^(2)+c^(2)a^(2)+a^(2)b^(2))`

Explanation

To find the area of triangle \( \Delta PQR \) given the midpoints of its sides, we can follow these steps:

### Step 1: Identify the midpoints and their coordinates
The midpoints of the sides of triangle \( PQR \) are given as:
- \( L(3a, 0, 0) \)
- \( M(0, 3b, 0) \)
- \( N(0, 0, 3c) \)

### Step 2: Determine the vectors for the sides of triangle \( LMN \)
We will calculate the vectors \( \overrightarrow{LM} \) and \( \overrightarrow{LN} \):
- \( \overrightarrow{LM} = M - L = (0 - 3a, 3b - 0, 0 - 0) = (-3a, 3b, 0) \)
- \( \overrightarrow{LN} = N - L = (0 - 3a, 0 - 0, 3c - 0) = (-3a, 0, 3c) \)

### Step 3: Calculate the cross product of the vectors
The area of triangle \( LMN \) can be found using the formula:
\[ \text{Area} = \frac{1}{2} \left| \overrightarrow{LM} \times \overrightarrow{LN} \right| \]
Calculating the cross product:
\[ \overrightarrow{LM} \times \overrightarrow{LN} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -3a & 3b & 0 \\ -3a & 0 & 3c \end{vmatrix} \]
Calculating the determinant:
\[ = \hat{i} \left( 3b \cdot 3c - 0 \cdot 0 \right) - \hat{j} \left( -3a \cdot 3c - 0 \cdot -3a \right) + \hat{k} \left( -3a \cdot 0 - 3b \cdot -3a \right) \]
\[ = 9bc \hat{i} + 9ac \hat{j} + 9ab \hat{k} \]

### Step 4: Calculate the magnitude of the cross product
The magnitude of the cross product is:
\[ \left| \overrightarrow{LM} \times \overrightarrow{LN} \right| = \sqrt{(9bc)^2 + (9ac)^2 + (9ab)^2} = 9 \sqrt{b^2c^2 + a^2c^2 + a^2b^2} \]

### Step 5: Calculate the area of triangle \( LMN \)
Now, substituting back into the area formula:
\[ \text{Area of } \Delta LMN = \frac{1}{2} \times 9 \sqrt{b^2c^2 + a^2c^2 + a^2b^2} = \frac{9}{2} \sqrt{b^2c^2 + a^2c^2 + a^2b^2} \]

### Step 6: Calculate the area of triangle \( PQR \)
Since the area of triangle \( PQR \) is four times the area of triangle \( LMN \):
\[ \text{Area of } \Delta PQR = 4 \times \text{Area of } \Delta LMN = 4 \times \frac{9}{2} \sqrt{b^2c^2 + a^2c^2 + a^2b^2} = 18 \sqrt{b^2c^2 + a^2c^2 + a^2b^2} \]

### Final Answer
The area of triangle \( PQR \) is:
\[ \text{Area of } \Delta PQR = 18 \sqrt{b^2c^2 + a^2c^2 + a^2b^2} \]

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