If each median of an equilateral triangle has a length of 12cm. Then what will be the value of In-radius (in cm.) of the equilateral triangle?

Options

A. 3
B. 4
C. 5
D. 6

Answer

To find the in-radius of an equilateral triangle given that each median has a length of 12 cm, we can follow these steps:

### Step 1: Understand the relationship between the median and the side length of an equilateral triangle.
The length of the median \( m \) of an equilateral triangle with side length \( a \) is given by the formula:
\[ m = \frac{\sqrt{3}}{2} a \]
Given that the median length is 12 cm, we can set up the equation:
\[ \frac{\sqrt{3}}{2} a = 12 \]

### Step 2: Solve for the side length \( a \).
To find \( a \), we can rearrange the equation:
\[ a = \frac{12 \times 2}{\sqrt{3}} = \frac{24}{\sqrt{3}} \]
To simplify this, we can multiply the numerator and denominator by \( \sqrt{3} \):
\[ a = \frac{24\sqrt{3}}{3} = 8\sqrt{3} \text{ cm} \]

### Step 3: Use the formula for the in-radius \( r \) of an equilateral triangle.
The in-radius \( r \) of an equilateral triangle can be calculated using the formula:
\[ r = \frac{a}{2\sqrt{3}} \]
Substituting the value of \( a \):
\[ r = \frac{8\sqrt{3}}{2\sqrt{3}} = \frac{8}{2} = 4 \text{ cm} \]

### Final Answer:
The in-radius of the equilateral triangle is \( 4 \) cm.

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