In `Delta ABC, angle B = 90^(@), AC = 29 cm and BC = 20 cm. ` Then `( 1 - sin A + cos A)/( 1 + sin A + cos A) ` is equal to :

Options

A. A)`3/8`
B. B)`3/7`
C. C)`1/2`
D. D)`1/4`

Answer

To solve the problem, we need to find the value of the expression \((1 - \sin A + \cos A) / (1 + \sin A + \cos A)\) in triangle \(ABC\) where \(\angle B = 90^\circ\), \(AC = 29 \, \text{cm}\), and \(BC = 20 \, \text{cm}\).

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
- Given that \(AC\) is the hypotenuse (29 cm) and \(BC\) is one of the legs (20 cm), we need to find the length of the other leg \(AB\).

2. **Use the Pythagorean theorem**:
- According to the Pythagorean theorem, \(AC^2 = AB^2 + BC^2\).
- Plugging in the values:
\[ 29^2 = AB^2 + 20^2 \]
\[ 841 = AB^2 + 400 \]
\[ AB^2 = 841 - 400 = 441 \]
\[ AB = \sqrt{441} = 21 \, \text{cm} \]

3. **Find \(\sin A\) and \(\cos A\)**:
- In triangle \(ABC\):
- \(\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{20}{29}\)
- \(\cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AB}{AC} = \frac{21}{29}\)

4. **Substitute \(\sin A\) and \(\cos A\) into the expression**:
- We need to calculate:
\[ \frac{1 - \sin A + \cos A}{1 + \sin A + \cos A} \]
- Substitute the values:
\[ \frac{1 - \frac{20}{29} + \frac{21}{29}}{1 + \frac{20}{29} + \frac{21}{29}} \]

5. **Simplify the numerator and denominator**:
- For the numerator:
\[ 1 - \frac{20}{29} + \frac{21}{29} = \frac{29}{29} - \frac{20}{29} + \frac{21}{29} = \frac{29 - 20 + 21}{29} = \frac{30}{29} \]
- For the denominator:
\[ 1 + \frac{20}{29} + \frac{21}{29} = \frac{29}{29} + \frac{20}{29} + \frac{21}{29} = \frac{29 + 20 + 21}{29} = \frac{70}{29} \]

6. **Final calculation**:
- Now, we can simplify the entire expression:
\[ \frac{\frac{30}{29}}{\frac{70}{29}} = \frac{30}{70} = \frac{3}{7} \]

### Conclusion:
The value of \(\frac{1 - \sin A + \cos A}{1 + \sin A + \cos A}\) is \(\frac{3}{7}\).