A circle of radius 10 cm has an equilateral triangle inscribed in it. The length of the perpendicular drawn from the centre to any side of the triangle is

Options

A. `2.5sqrt(3)` cm
B. 5 cm
C. `10sqrt(3)` cm
D. None of these

Answer

To find the length of the perpendicular drawn from the center of a circle to any side of an inscribed equilateral triangle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the radius of the circle**:
The radius (R) of the circle is given as 10 cm.

2. **Understand the relationship between the radius and the side of the equilateral triangle**:
For an equilateral triangle inscribed in a circle, the relationship between the radius (R) of the circumscribed circle and the side length (A) of the triangle is given by the formula:
\[ R = \frac{A}{\sqrt{3}} \]

3. **Rearrange the formula to find the side length (A)**:
From the formula, we can express the side length (A) in terms of the radius (R):
\[ A = R \cdot \sqrt{3} \]
Substituting R = 10 cm:
\[ A = 10 \cdot \sqrt{3} \text{ cm} \]

4. **Determine the length of the perpendicular from the center to a side**:
The length of the perpendicular (h) from the center of the circle to any side of the triangle can be calculated using the formula:
\[ h = \frac{A \cdot \sqrt{3}}{2} \]
Substituting A = \(10 \cdot \sqrt{3}\):
\[ h = \frac{(10 \cdot \sqrt{3}) \cdot \sqrt{3}}{2} \]
Simplifying this:
\[ h = \frac{10 \cdot 3}{2} = \frac{30}{2} = 15 \text{ cm} \]

5. **Find the perpendicular distance from the center to the side**:
However, we need to find the perpendicular distance from the center to the side of the triangle. The correct formula for the height (h) from the center of the triangle to a side is:
\[ h = \frac{R \cdot \sqrt{3}}{2} \]
Substituting R = 10 cm:
\[ h = \frac{10 \cdot \sqrt{3}}{2} = 5\sqrt{3} \text{ cm} \]

### Final Answer:
The length of the perpendicular drawn from the center to any side of the triangle is \(5\sqrt{3}\) cm.