In `DeltaABC," if " angleC=90^(@)," then " (a+c)/(b)+(b+c)/(a)` is equal to :

Options

A. `(c )/(r )`
B. `(1)/(2Rr)`
C. 2
D. `(R )/(r )`

Answer

To solve the problem, we need to find the value of the expression \(\frac{a+c}{b} + \frac{b+c}{a}\) given that \(\angle C = 90^\circ\) in triangle \(ABC\).

### Step-by-Step Solution:

1. **Understanding the Triangle**:
Since \(\angle C = 90^\circ\), triangle \(ABC\) is a right triangle with \(C\) as the right angle. Here, \(a\), \(b\), and \(c\) represent the lengths of the sides opposite to angles \(A\), \(B\), and \(C\) respectively.

2. **Using Pythagorean Theorem**:
From the Pythagorean theorem, we know that:
\[ a^2 + b^2 = c^2 \]

3. **Rewriting the Expression**:
We need to simplify the expression:
\[ \frac{a+c}{b} + \frac{b+c}{a} \]
We can find a common denominator, which is \(ab\):
\[ = \frac{(a+c)a + (b+c)b}{ab} \]

4. **Expanding the Numerator**:
Expanding the numerator gives:
\[ = \frac{a^2 + ac + b^2 + bc}{ab} \]

5. **Substituting Pythagorean Identity**:
We can substitute \(a^2 + b^2\) with \(c^2\):
\[ = \frac{c^2 + ac + bc}{ab} \]

6. **Factoring the Numerator**:
The numerator can be factored:
\[ = \frac{c^2 + c(a + b)}{ab} \]
This can be rewritten as:
\[ = \frac{c(c + a + b)}{ab} \]

7. **Using the Semi-Perimeter**:
The semi-perimeter \(s\) of triangle \(ABC\) is given by:
\[ s = \frac{a + b + c}{2} \]
Therefore, \(2s = a + b + c\).

8. **Final Expression**:
Substituting \(2s\) into our expression gives:
\[ = \frac{c(2s)}{ab} \]

9. **Using the Circumradius**:
The circumradius \(R\) of triangle \(ABC\) is given by:
\[ R = \frac{abc}{4\Delta} \]
where \(\Delta\) is the area of triangle \(ABC\). For a right triangle, \(\Delta = \frac{1}{2}ab\), hence:
\[ R = \frac{abc}{2ab} = \frac{c}{2} \]

10. **Substituting for \(R\)**:
Therefore, we can express \(c\) in terms of \(R\):
\[ c = 2R \]

11. **Final Result**:
Substituting \(c\) back into our expression gives:
\[ = \frac{2R(2s)}{ab} = \frac{4Rs}{ab} \]

12. **Conclusion**:
The final value of the expression \(\frac{a+c}{b} + \frac{b+c}{a}\) simplifies to:
\[ = \frac{c}{r} \]
where \(r\) is the radius of the inscribed circle.

### Final Answer:
\[ \frac{c}{r} \]