If the radius of the incircle of a triangle withits sides 5k, 6k and 5k is 6, then k is equal to

Options

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

Answer

To find the value of \( k \) given that the radius of the incircle of a triangle with sides \( 5k, 6k, \) and \( 5k \) is \( 6 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
- Let the sides of the triangle be \( a = 5k \), \( b = 6k \), and \( c = 5k \).

2. **Calculate the semi-perimeter \( s \)**:
- The semi-perimeter \( s \) is given by the formula:
\[ s = \frac{a + b + c}{2} \]
- Substituting the values:
\[ s = \frac{5k + 6k + 5k}{2} = \frac{16k}{2} = 8k \]

3. **Use the formula for the radius of the incircle**:
- The radius \( r \) of the incircle is given by:
\[ r = \frac{\Delta}{s} \]
- Where \( \Delta \) is the area of the triangle.

4. **Calculate the area \( \Delta \) using Heron's formula**:
- Heron's formula states:
\[ \Delta = \sqrt{s(s-a)(s-b)(s-c)} \]
- Calculate \( s - a, s - b, \) and \( s - c \):
- \( s - a = 8k - 5k = 3k \)
- \( s - b = 8k - 6k = 2k \)
- \( s - c = 8k - 5k = 3k \)

5. **Substitute into Heron's formula**:
- Now substituting into Heron's formula:
\[ \Delta = \sqrt{8k \cdot 3k \cdot 2k \cdot 3k} \]
- Simplifying this:
\[ \Delta = \sqrt{48k^4} = 12k^2 \]

6. **Set up the equation using the given radius**:
- We know \( r = 6 \), so:
\[ 6 = \frac{12k^2}{8k} \]
- Simplifying the right-hand side:
\[ 6 = \frac{12k^2}{8k} = \frac{3k}{2} \]

7. **Solve for \( k \)**:
- Cross-multiplying gives:
\[ 6 \cdot 2 = 3k \implies 12 = 3k \implies k = \frac{12}{3} = 4 \]

### Final Answer:
Thus, the value of \( k \) is \( 4 \).

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