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

Options

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

Answer

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

### Step 1: Understand the relationship between the perpendiculars and the height of the triangle.
The sum of the lengths of the perpendiculars drawn from any point inside an equilateral triangle to its three sides is equal to the height of the triangle.

Given:
- Perpendicular to side AB = 6 cm
- Perpendicular to side BC = 7 cm
- Perpendicular to side CA = 8 cm

### Step 2: Calculate the total height of the triangle.
To find the height (H) of the triangle, we add the lengths of the three perpendiculars:
\[ H = 6 \, \text{cm} + 7 \, \text{cm} + 8 \, \text{cm} = 21 \, \text{cm} \]

### Step 3: Use the relationship between the height and the side of the triangle.
For an equilateral triangle, the relationship between the side length (A) and the height (H) is given by the formula:
\[ H = \frac{A \sqrt{3}}{2} \]

### Step 4: Rearrange the formula to find the side length.
We can rearrange the formula to solve for A:
\[ A = \frac{2H}{\sqrt{3}} \]

### Step 5: Substitute the height into the formula.
Now, substituting the value of H into the equation:
\[ A = \frac{2 \times 21}{\sqrt{3}} = \frac{42}{\sqrt{3}} \]

### Step 6: Simplify the expression.
To simplify \( \frac{42}{\sqrt{3}} \), we can multiply the numerator and the denominator by \( \sqrt{3} \):
\[ A = \frac{42 \sqrt{3}}{3} = 14 \sqrt{3} \, \text{cm} \]

### Final Answer:
The length of the side of the triangle is \( 14 \sqrt{3} \, \text{cm} \).

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