A point within an equilateral triangle, where perimeter is 18 m is 1 m from one side and 2 m from another side. Its distance from the third side is :

Options

A. `3 sqrt(3) + 3`
B. `3 sqrt(3) - 3`
C. `3 - sqrt(3)`
D. `3 + sqrt(3)`

Answer

To solve the problem, we need to find the distance from a point inside an equilateral triangle to the third side, given the distances to the other two sides.

### Step-by-Step Solution:

1. **Understand the Problem**: We have an equilateral triangle with a perimeter of 18 m. This means each side of the triangle is \( \frac{18}{3} = 6 \) m. We need to find the distance from a point inside the triangle to the third side, given that the distances to the other two sides are 1 m and 2 m.

2. **Assign Variables**: Let the distance from the point to the third side be \( x \) m. We know the distances to the other two sides are 1 m and 2 m.

3. **Calculate the Area of the Triangle**: The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} \times \text{side}^2 \]
Substituting the side length of 6 m:
\[ A = \frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \text{ m}^2 \]

4. **Calculate the Area Using Distances**: The area of the triangle can also be expressed as the sum of the areas of three smaller triangles formed by the point and the sides of the triangle. The area of each triangle can be calculated as:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
- For the side with distance 1 m:
\[ A_1 = \frac{1}{2} \times 6 \times 1 = 3 \text{ m}^2 \]
- For the side with distance 2 m:
\[ A_2 = \frac{1}{2} \times 6 \times 2 = 6 \text{ m}^2 \]
- For the side with distance \( x \) m:
\[ A_3 = \frac{1}{2} \times 6 \times x = 3x \text{ m}^2 \]

5. **Set Up the Equation**: The total area of the triangle can be expressed as:
\[ A = A_1 + A_2 + A_3 \]
Substituting the areas we calculated:
\[ 9\sqrt{3} = 3 + 6 + 3x \]
Simplifying gives:
\[ 9\sqrt{3} = 9 + 3x \]

6. **Solve for \( x \)**: Rearranging the equation to isolate \( x \):
\[ 3x = 9\sqrt{3} - 9 \]
Dividing both sides by 3:
\[ x = 3\sqrt{3} - 3 \]

7. **Conclusion**: The distance from the point to the third side is \( x = 3\sqrt{3} - 3 \) m.