If the angles of a triangle are in the ratio `1:3:5` then the angle greatest angle is

Options

A. `(5pi)/(9)`
B. `(2pi)/(9)`
C. `(7pi)/(9)`
D. `(11pi)/(9)`

Answer

To find the greatest angle of a triangle whose angles are in the ratio \(1:3:5\), we can follow these steps:

### Step 1: Understand the ratio of the angles
The angles of the triangle are given in the ratio \(1:3:5\). Let's denote the angles as:
- First angle = \(x\)
- Second angle = \(3x\)
- Third angle = \(5x\)

### Step 2: Set up the equation for the sum of angles in a triangle
The sum of the angles in any triangle is \(180^\circ\) (or \(\pi\) radians). Therefore, we can write the equation:
\[ x + 3x + 5x = 180^\circ \]

### Step 3: Simplify the equation
Combine the terms on the left side:
\[ 9x = 180^\circ \]

### Step 4: Solve for \(x\)
To find \(x\), divide both sides by \(9\):
\[ x = \frac{180^\circ}{9} = 20^\circ \]

### Step 5: Calculate each angle
Now, we can find each angle:
- First angle = \(x = 20^\circ\)
- Second angle = \(3x = 3 \times 20^\circ = 60^\circ\)
- Third angle = \(5x = 5 \times 20^\circ = 100^\circ\)

### Step 6: Identify the greatest angle
From the calculated angles, the greatest angle is:
\[ 100^\circ \]

### Step 7: Convert to radians
To express the greatest angle in radians, we convert \(100^\circ\) to radians:
\[ 100^\circ = \frac{100 \times \pi}{180} = \frac{5\pi}{9} \]

### Conclusion
Thus, the greatest angle in the triangle is \(\frac{5\pi}{9}\).