Statement I If the sides of a triangle are 13, 14 15 then the radius of in circle =4
Statement II In `a DeltaABC, Delta = sqrt(s (s-a) (s-b) (s-c))`where `s=(a+b+c)/(2) and r =(Delta)/(s)`

Options

A. Statement I is True, Statement II is True, Statement II is a correct explanation for Statement I.
B. Statement I is True, Statement II is True, Statement II is NOT a correct explanation for Statement I.
C. Statement I is True, Statement II is False.
D. Statement I is False, Statement II is True.

Answer

To solve the problem, we need to verify the two statements regarding the triangle with sides 13, 14, and 15.

### Step-by-Step Solution:

1. **Identify the sides of the triangle:**
Let the sides of the triangle be:
- \( a = 13 \)
- \( b = 14 \)
- \( c = 15 \)

2. **Calculate the semi-perimeter (s):**
The semi-perimeter \( s \) is given by the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 \]

3. **Calculate the area (Δ) using Heron's formula:**
Heron's formula states:
\[ \Delta = \sqrt{s(s-a)(s-b)(s-c)} \]
Now substitute the values:
\[ \Delta = \sqrt{21(21-13)(21-14)(21-15)} \]
Simplifying further:
\[ \Delta = \sqrt{21 \times 8 \times 7 \times 6} \]

4. **Simplify the expression under the square root:**
Calculate each term:
- \( s - a = 21 - 13 = 8 \)
- \( s - b = 21 - 14 = 7 \)
- \( s - c = 21 - 15 = 6 \)

Now, we have:
\[ \Delta = \sqrt{21 \times 8 \times 7 \times 6} \]

5. **Factor the expression:**
We can factor it as follows:
\[ 21 = 7 \times 3, \quad 8 = 2^3, \quad 6 = 2 \times 3 \]
Thus:
\[ \Delta = \sqrt{(7 \times 3) \times (2^3) \times 7 \times (2 \times 3)} = \sqrt{7^2 \times 3^2 \times 2^4} \]

6. **Calculate the square root:**
Taking the square root:
\[ \Delta = 7 \times 3 \times 2^2 = 7 \times 3 \times 4 = 84 \]

7. **Calculate the radius of the incircle (r):**
The radius \( r \) of the incircle is given by:
\[ r = \frac{\Delta}{s} \]
Substituting the values:
\[ r = \frac{84}{21} = 4 \]

### Conclusion:
- **Statement I** is true: The radius of the incircle is indeed 4.
- **Statement II** is also true as it correctly describes the relationship between the area, semi-perimeter, and the radius of the incircle.

Thus, both statements are correct.