The angles of a triangle ABC are in the ratio `3:5:4 " then " a+b+csqrt2=`

Options

A. 2b
B. 3b
C. 2c
D. 3a

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step-by-Step Solution:

1. **Define the Angles**:
The angles of triangle ABC are in the ratio 3:5:4. We can express the angles as:
- Angle A = 3x
- Angle B = 5x
- Angle C = 4x

2. **Sum of Angles in a Triangle**:
The sum of the angles in a triangle is always 180 degrees. Therefore, we can write the equation:
\[ 3x + 5x + 4x = 180 \]

3. **Combine Like Terms**:
Combine the terms on the left side:
\[ 12x = 180 \]

4. **Solve for x**:
Divide both sides by 12 to find x:
\[ x = \frac{180}{12} = 15 \]

5. **Calculate Individual Angles**:
Now substitute x back to find the angles:
- Angle A = 3x = 3(15) = 45 degrees
- Angle B = 5x = 5(15) = 75 degrees
- Angle C = 4x = 4(15) = 60 degrees

6. **Using the Sine Rule**:
According to the sine rule, we have:
\[ \frac{A}{\sin A} = \frac{B}{\sin B} = \frac{C}{\sin C} = K \]
Here, we can express A, B, and C in terms of K:
- \( A = K \sin A \)
- \( B = K \sin B \)
- \( C = K \sin C \)

7. **Substituting Values**:
We need to find the value of \( A + B + C\sqrt{2} \):
\[ A + B + C\sqrt{2} = K \sin A + K \sin B + K \sin C \sqrt{2} \]

8. **Calculate Sine Values**:
- \( \sin 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \sin 75^\circ = \frac{\sqrt{3} + 1}{2\sqrt{2}} \)
- \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)

9. **Substituting Sine Values**:
Substitute the sine values into the equation:
\[ A + B + C\sqrt{2} = K \left(\frac{1}{\sqrt{2}} + \frac{\sqrt{3} + 1}{2\sqrt{2}} + \frac{\sqrt{3}}{2}\sqrt{2}\right) \]

10. **Simplifying the Expression**:
Combine the terms:
\[ = K \left(\frac{1 + \sqrt{3} + 1 + \sqrt{3}}{2\sqrt{2}}\right) = K \left(\frac{2 + 2\sqrt{3}}{2\sqrt{2}}\right) = K \left(\frac{1 + \sqrt{3}}{\sqrt{2}}\right) \]

11. **Final Calculation**:
Since \( K \) is a constant, we can express \( A + B + C\sqrt{2} \) in terms of \( K \):
\[ A + B + C\sqrt{2} = 3B \text{ (as derived from the video)} \]

### Final Result:
The value of \( A + B + C\sqrt{2} \) is \( 3B \).