A point 'P' is an arbitrary interior point of an equilateral triangle of side 4. If x, y, z are the distances of 'P' from sides of the triangle then the value of `(x+y+z)^(2)` is equal to -

Options

A. 3
B. 12
C. 18
D. 48

Answer

To solve the problem, we need to find the value of \((x + y + z)^2\), where \(x\), \(y\), and \(z\) are the distances from an arbitrary point \(P\) inside an equilateral triangle with side length 4 to its sides.

### Step 1: Understand the Geometry
We have an equilateral triangle \(ABC\) with vertices \(A\), \(B\), and \(C\) and side length \(4\). Let \(P\) be an arbitrary point inside the triangle. The distances from point \(P\) to the sides \(BC\), \(CA\), and \(AB\) are denoted as \(x\), \(y\), and \(z\) respectively.

### Step 2: Area of the Triangle
The area \(A\) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
where \(a\) is the length of a side. Here, \(a = 4\):
\[ A = \frac{\sqrt{3}}{4} \times 4^2 = \frac{\sqrt{3}}{4} \times 16 = 4\sqrt{3} \]

### Step 3: Area Using Distances
The area of triangle \(ABC\) can also be expressed as the sum of the areas of the three smaller triangles formed by point \(P\) and the sides of the triangle. The area of triangle \(APB\) is:
\[ \text{Area}_{APB} = \frac{1}{2} \times AB \times z = \frac{1}{2} \times 4 \times z = 2z \]
The area of triangle \(BPC\) is:
\[ \text{Area}_{BPC} = \frac{1}{2} \times BC \times x = \frac{1}{2} \times 4 \times x = 2x \]
The area of triangle \(APC\) is:
\[ \text{Area}_{APC} = \frac{1}{2} \times AC \times y = \frac{1}{2} \times 4 \times y = 2y \]

### Step 4: Total Area
Thus, the total area of triangle \(ABC\) can be expressed as:
\[ \text{Area}_{ABC} = \text{Area}_{APB} + \text{Area}_{BPC} + \text{Area}_{APC} = 2x + 2y + 2z \]
This gives us:
\[ \text{Area}_{ABC} = 2(x + y + z) \]

### Step 5: Equating Areas
Now we can equate the two expressions for the area of triangle \(ABC\):
\[ 4\sqrt{3} = 2(x + y + z) \]
Dividing both sides by 2:
\[ 2\sqrt{3} = x + y + z \]

### Step 6: Finding \((x + y + z)^2\)
Now we need to find \((x + y + z)^2\):
\[ (x + y + z)^2 = (2\sqrt{3})^2 = 4 \times 3 = 12 \]

### Final Answer
Thus, the value of \((x + y + z)^2\) is:
\[ \boxed{12} \]