The angles of a triangle are in the ratio of `1:2:3`. What will be the radian measure of the largest angle of the triangle ?

Options

A. `pi/2`
B. `pi/3`
C. `pi/4`
D. `(2pi)/3`

Answer

To find the radian measure of the largest angle of a triangle where the angles are in the ratio of 1:2:3, we can follow these steps:

### Step 1: Set up the angles based on the ratio
Let the angles of the triangle be represented as:
- First angle = \( x \)
- Second angle = \( 2x \)
- Third angle = \( 3x \)

### Step 2: Write the equation for the sum of angles in a triangle
The sum of the angles in a triangle is always \( 180^\circ \). Therefore, we can write the equation:
\[ x + 2x + 3x = 180^\circ \]

### Step 3: Combine like terms
Combine the terms on the left side:
\[ 6x = 180^\circ \]

### Step 4: Solve for \( x \)
To find the value of \( x \), divide both sides by 6:
\[ x = \frac{180^\circ}{6} = 30^\circ \]

### Step 5: Calculate the angles
Now we can find the measures of the three angles:
- First angle = \( x = 30^\circ \)
- Second angle = \( 2x = 2 \times 30^\circ = 60^\circ \)
- Third angle = \( 3x = 3 \times 30^\circ = 90^\circ \)

### Step 6: Identify the largest angle
The largest angle among \( 30^\circ, 60^\circ, \) and \( 90^\circ \) is \( 90^\circ \).

### Step 7: Convert the largest angle to radians
To convert degrees to radians, we use the conversion factor \( \frac{\pi \text{ radians}}{180^\circ} \):
\[ 90^\circ \times \frac{\pi}{180} = \frac{\pi}{2} \text{ radians} \]

### Final Answer
The radian measure of the largest angle of the triangle is:
\[ \frac{\pi}{2} \text{ radians} \]

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