The in-radius of an equilateral triangle is of length 3 cm. Then the length of each of its medians is

Options

A. 12 cm
B. `9/2` cm
C. 4 cm
D. 9 cm

Answer

To find the length of each median of an equilateral triangle given that the in-radius is 3 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the In-radius**:
The in-radius (r) of an equilateral triangle is the radius of the circle inscribed within the triangle. For our case, r = 3 cm.

2. **Finding the Side Length (s)**:
The formula for the in-radius (r) of an equilateral triangle in terms of its side length (s) is:
\[ r = \frac{s \sqrt{3}}{6} \]
Rearranging this formula to find s gives:
\[ s = \frac{6r}{\sqrt{3}} \]
Substituting r = 3 cm:
\[ s = \frac{6 \times 3}{\sqrt{3}} = \frac{18}{\sqrt{3}} = 6\sqrt{3} \text{ cm} \]

3. **Finding the Length of the Median (m)**:
The formula for the length of the median (m) of an equilateral triangle in terms of its side length (s) is:
\[ m = \frac{s \sqrt{3}}{2} \]
Substituting s = 6√3 cm:
\[ m = \frac{6\sqrt{3} \times \sqrt{3}}{2} = \frac{6 \times 3}{2} = \frac{18}{2} = 9 \text{ cm} \]

4. **Conclusion**:
Therefore, the length of each median of the equilateral triangle is **9 cm**.

### Summary:
The length of each median of the equilateral triangle is 9 cm.

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