Statement I In a `Delta ABC, if a lt b lt c` and r is inradius and `r_(1) , r_(2) , r_(3)` are the exradii opposite to angle A,B,C respectively, then `r lt r_(1) lt r_(2) lt r_(3).`
Statement II For, `DeltaABC r_(1)r_(2)+r_(2)r_(3)+r_(3)r_(1)=(r_(1)r_(2)r_(3))/(r)`

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 given problem, we need to analyze the two statements provided regarding triangle \( \Delta ABC \) with sides \( a < b < c \), where \( r \) is the inradius and \( r_1, r_2, r_3 \) are the exradii opposite to angles \( A, B, C \) respectively.

### Step 1: Understanding the Inradius and Exradii
The inradius \( r \) and exradii \( r_1, r_2, r_3 \) can be expressed using the area \( \Delta \) of the triangle and the semi-perimeter \( s \):
- \( r = \frac{\Delta}{s} \)
- \( r_1 = \frac{\Delta}{s-a} \)
- \( r_2 = \frac{\Delta}{s-b} \)
- \( r_3 = \frac{\Delta}{s-c} \)

### Step 2: Comparing the Inradius and Exradii
Since \( a < b < c \), it follows that:
- \( s - a > s - b > s - c \)

This means that the denominators for \( r_1, r_2, r_3 \) are in descending order:
- \( s - a > s - b > s - c \)

### Step 3: Establishing the Inequalities
As the denominators decrease, the values of the exradii increase:
- \( r_1 = \frac{\Delta}{s-a} > r_2 = \frac{\Delta}{s-b} > r_3 = \frac{\Delta}{s-c} \)

Thus, we can conclude:
- \( r < r_1 < r_2 < r_3 \)

This confirms Statement I: \( r < r_1 < r_2 < r_3 \).

### Step 4: Verifying Statement II
Now, we need to verify Statement II:
\[ r_1 r_2 + r_2 r_3 + r_3 r_1 = \frac{r_1 r_2 r_3}{r} \]

Substituting the expressions for \( r_1, r_2, r_3 \):
- Left-hand side:
\[ r_1 r_2 + r_2 r_3 + r_3 r_1 = \frac{\Delta^2}{(s-a)(s-b)} + \frac{\Delta^2}{(s-b)(s-c)} + \frac{\Delta^2}{(s-c)(s-a)} \]
Taking the common denominator \( (s-a)(s-b)(s-c) \) and simplifying gives us a formula involving \( s \) and the area \( \Delta \).

- Right-hand side:
\[ \frac{r_1 r_2 r_3}{r} = \frac{\left(\frac{\Delta}{s-a}\right)\left(\frac{\Delta}{s-b}\right)\left(\frac{\Delta}{s-c}\right)}{\frac{\Delta}{s}} = \frac{\Delta^2 s}{(s-a)(s-b)(s-c)} \]

### Step 5: Conclusion
After simplifying both sides, we find that they are indeed equal, confirming Statement II is also true.

### Final Result
Both statements are true, but Statement II does not serve as a correct explanation for Statement I. Thus, the conclusion is that both statements are true, but Statement II is not a correct explanation of Statement I.