From a point within an equilateral triangle, perpendiculars drawn to the three sides are 6 cm, 7 cm and 8 cm respectively. The length of the side of the triangle is

Options

A. 7 cm
B. 10.5 cm
C. `14 sqrt(3)` cm
D. `(14 sqrt(3))/(3)` cm

Answer

To find the length of the side of the equilateral triangle given the perpendiculars from a point within the triangle, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have an equilateral triangle ABC and a point O inside it. We are given the lengths of the perpendiculars from point O to the sides of the triangle: \( h_a = 6 \, \text{cm} \), \( h_b = 7 \, \text{cm} \), and \( h_c = 8 \, \text{cm} \).

2. **Formula for Area of Equilateral Triangle**: The area \( A \) of an equilateral triangle with side length \( a \) is given by:
\[ A = \frac{\sqrt{3}}{4} a^2 \]

3. **Area Calculation Using Perpendiculars**: The area of triangle ABC can also be expressed using the perpendiculars:
\[ A = \frac{1}{2} \times a \times h_a + \frac{1}{2} \times a \times h_b + \frac{1}{2} \times a \times h_c \]
This simplifies to:
\[ A = \frac{1}{2} a (h_a + h_b + h_c) \]

4. **Substituting Values**: Substitute the values of the perpendiculars:
\[ A = \frac{1}{2} a (6 + 7 + 8) = \frac{1}{2} a \times 21 = \frac{21a}{2} \]

5. **Equating the Two Area Expressions**: Now, we can set the two expressions for area equal to each other:
\[ \frac{\sqrt{3}}{4} a^2 = \frac{21a}{2} \]

6. **Eliminating \( a \)**: To eliminate \( a \) from both sides, we can multiply both sides by 4:
\[ \sqrt{3} a^2 = 42a \]

7. **Rearranging the Equation**: Rearranging gives us:
\[ \sqrt{3} a^2 - 42a = 0 \]

8. **Factoring Out \( a \)**: Factoring out \( a \):
\[ a(\sqrt{3} a - 42) = 0 \]
This gives us two solutions: \( a = 0 \) or \( \sqrt{3} a - 42 = 0 \).

9. **Solving for \( a \)**: Solving \( \sqrt{3} a - 42 = 0 \):
\[ \sqrt{3} a = 42 \implies a = \frac{42}{\sqrt{3}} = 14\sqrt{3} \, \text{cm} \]

10. **Final Answer**: The length of the side of the triangle is:
\[ a = 14\sqrt{3} \, \text{cm} \]