The perpendicular distances of the three sides of a equilateral triangle from an interior point are 6 cms, 9 cms and 12 cms respectively. Find the perimeter of the triangle

Options

A. 36 cm
B. ` 54 sqrt"" 3 cm `
C. ` 18 sqrt"" 3 ` cm
D. none of these

Answer

To find the perimeter of the equilateral triangle given the perpendicular distances from an interior point to its sides, we can follow these steps:

### Step 1: Understand the Problem
We have an equilateral triangle with three sides, and the perpendicular distances from an interior point to these sides are given as:
- \( P_1 = 6 \) cm
- \( P_2 = 9 \) cm
- \( P_3 = 12 \) cm

### Step 2: Use the Property of Perpendicular Distances
For any triangle, the sum of the perpendicular distances from an interior point to the three sides is equal to the height of the triangle. Therefore, we can write:
\[ P_1 + P_2 + P_3 = \text{Height of the triangle} \]
Substituting the values:
\[ 6 + 9 + 12 = \text{Height of the triangle} \]
Calculating the sum:
\[ 27 = \text{Height of the triangle} \]

### Step 3: Relate Height to Side Length
For an equilateral triangle with side length \( A \), the height \( h \) can be expressed as:
\[ h = \frac{\sqrt{3}}{2} A \]
Setting this equal to the height we found:
\[ 27 = \frac{\sqrt{3}}{2} A \]

### Step 4: Solve for Side Length \( A \)
To find \( A \), we rearrange the equation:
\[ A = \frac{27 \times 2}{\sqrt{3}} = \frac{54}{\sqrt{3}} \]
To rationalize the denominator:
\[ A = \frac{54 \sqrt{3}}{3} = 18 \sqrt{3} \]

### Step 5: Calculate the Perimeter
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3A \]
Substituting the value of \( A \):
\[ P = 3 \times 18 \sqrt{3} = 54 \sqrt{3} \]

### Final Answer
Thus, the perimeter of the equilateral triangle is:
\[ \boxed{54 \sqrt{3} \text{ cm}} \]

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