From a point in the interior of an equilateral triangle the perpendiculr distances of the sides are `sqrt(3)` cm, `2 sqrt(3)` cm and `5 sqrt(3)`. What is the perimeter (in cm) of the triangle ?

Answer

To find the perimeter of the equilateral triangle given the perpendicular distances from a point inside the triangle to its sides, we will follow these steps:

### Step 1: Understand the Problem
We have an equilateral triangle ABC, and from a point O inside the triangle, the perpendicular distances to the sides are given as:
- OD = √3 cm
- OE = 2√3 cm
- OF = 5√3 cm

### Step 2: Let the Side Length of the Triangle be A
Let the side length of the equilateral triangle ABC be A cm. Since all sides of an equilateral triangle are equal, we have:
- AB = BC = CA = A

### Step 3: Calculate the Area of the Triangle
The area of an equilateral triangle can be calculated using the formula:
\[ \text{Area} = \frac{\sqrt{3}}{4} A^2 \]

### Step 4: Calculate the Area Using Perpendicular Distances
We can also calculate the area of triangle ABC using the perpendicular distances from point O to the sides:
- Area of triangle BOC = \( \frac{1}{2} \times A \times OD = \frac{1}{2} \times A \times \sqrt{3} \)
- Area of triangle COA = \( \frac{1}{2} \times A \times OE = \frac{1}{2} \times A \times 2\sqrt{3} \)
- Area of triangle AOB = \( \frac{1}{2} \times A \times OF = \frac{1}{2} \times A \times 5\sqrt{3} \)

### Step 5: Sum the Areas
Now, we can sum the areas calculated using the perpendicular distances:
\[ \text{Total Area} = \frac{1}{2} A \sqrt{3} + \frac{1}{2} A \cdot 2\sqrt{3} + \frac{1}{2} A \cdot 5\sqrt{3} \]
\[ = \frac{1}{2} A (\sqrt{3} + 2\sqrt{3} + 5\sqrt{3}) = \frac{1}{2} A (8\sqrt{3}) = 4\sqrt{3} A \]

### Step 6: Set the Two Area Expressions Equal
Now we set the area calculated from the side length equal to the area calculated from the perpendicular distances:
\[ \frac{\sqrt{3}}{4} A^2 = 4\sqrt{3} A \]

### Step 7: Solve for A
To solve for A, we can multiply both sides by 4 to eliminate the fraction:
\[ \sqrt{3} A^2 = 16\sqrt{3} A \]
Now, divide both sides by \( \sqrt{3} \) (assuming A ≠ 0):
\[ A^2 = 16 A \]
Rearranging gives:
\[ A^2 - 16 A = 0 \]
Factoring out A:
\[ A(A - 16) = 0 \]
Thus, \( A = 0 \) or \( A = 16 \). Since A cannot be zero, we have:
\[ A = 16 \text{ cm} \]

### Step 8: Calculate the Perimeter
The perimeter of an equilateral triangle is given by:
\[ \text{Perimeter} = 3 \times A = 3 \times 16 = 48 \text{ cm} \]

### Final Answer
The perimeter of the triangle is **48 cm**.

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