By decreasing `15^(@)` of eqch angle of a triangle the ratios of their angles are 2:3:5 the radian meausre of greatest angle is

Options

A. `(pi)/(12)`
B. `(pi)/(24)`
C. `(5pi)/(24)`
D. `(11pi)/(24)`

Answer

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

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find the greatest angle of a triangle after decreasing each angle by 15 degrees. The remaining angles are in the ratio of 2:3:5.

2. **Calculate the Total Decrease**: Since there are three angles in the triangle and each angle is decreased by 15 degrees, the total decrease in degrees is:
\[ \text{Total decrease} = 3 \times 15 = 45 \text{ degrees} \]

3. **Calculate the Remaining Sum of Angles**: The sum of angles in a triangle is always 180 degrees. After the decrease, the remaining sum of angles is:
\[ \text{Remaining sum} = 180 - 45 = 135 \text{ degrees} \]

4. **Setting Up the Ratio**: Let the angles of the triangle be represented as \(2x\), \(3x\), and \(5x\) based on the given ratio of 2:3:5.

5. **Equation for the Remaining Angles**: The sum of the angles in terms of \(x\) is:
\[ 2x + 3x + 5x = 135 \]
Simplifying this gives:
\[ 10x = 135 \]

6. **Solving for \(x\)**: To find \(x\), divide both sides by 10:
\[ x = \frac{135}{10} = 13.5 \text{ degrees} \]

7. **Finding the Greatest Angle**: The greatest angle corresponds to the largest ratio part, which is \(5x\):
\[ \text{Greatest angle} = 5x = 5 \times 13.5 = 67.5 \text{ degrees} \]

8. **Adding the Decrease Back**: Since we need the original angle before the decrease, we add back the 15 degrees:
\[ \text{Original greatest angle} = 67.5 + 15 = 82.5 \text{ degrees} \]

9. **Convert to Radians**: To convert degrees to radians, use the formula:
\[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]
Thus,
\[ 82.5 \times \frac{\pi}{180} = \frac{82.5\pi}{180} = \frac{11\pi}{24} \]

### Final Answer:
The radian measure of the greatest angle is:
\[ \frac{11\pi}{24} \]