Perpendiculars are drawn on the side of an equilateral triangle from any point within the triangle. If the lengths of these perpendiculars be 6 cm, 7 cm and 9 cm, then the length of a side of the triangle is :

Options

A. `44/(3)sqrt(3)` cm
B. `11/(3)sqrt(3)` cm
C. `22/(3)sqrt(3)` cm
D. `11sqrt(3)` cm

Answer

To find the length of a side of the equilateral triangle given the lengths of the perpendiculars dropped from a point inside the triangle, we can use the property that the sum of the lengths of the perpendiculars from any point inside an equilateral triangle to its sides is equal to the height of the triangle.

### Step-by-Step Solution:

1. **Identify the lengths of the perpendiculars**:
- Let the lengths of the perpendiculars from the point inside the triangle to the sides be \( p_1 = 6 \, \text{cm} \), \( p_2 = 7 \, \text{cm} \), and \( p_3 = 9 \, \text{cm} \).

2. **Calculate the sum of the perpendiculars**:
\[ \text{Sum of perpendiculars} = p_1 + p_2 + p_3 = 6 + 7 + 9 = 22 \, \text{cm} \]

3. **Relate the sum of the perpendiculars to the height of the triangle**:
- The height \( h \) of an equilateral triangle can be expressed in terms of the side length \( a \) using the formula:
\[ h = \frac{\sqrt{3}}{2} a \]

4. **Set the sum of the perpendiculars equal to the height**:
\[ 22 = \frac{\sqrt{3}}{2} a \]

5. **Solve for the side length \( a \)**:
- Rearranging the equation gives:
\[ a = \frac{22 \times 2}{\sqrt{3}} = \frac{44}{\sqrt{3}} \]
- To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{3} \):
\[ a = \frac{44 \sqrt{3}}{3} \]

6. **Calculate the approximate value of \( a \)**:
- Using \( \sqrt{3} \approx 1.732 \):
\[ a \approx \frac{44 \times 1.732}{3} \approx \frac{76.168}{3} \approx 25.39 \, \text{cm} \]

Thus, the length of a side of the equilateral triangle is approximately \( 25.39 \, \text{cm} \).