If the angles of a triangle are in the ratio `1:4:7`, then the value of the largest angle is :

Options

A. `135^(@)`
B. `84^(@)`
C. `105^(@)`
D. none of these

Answer

To solve the problem of finding the largest angle in a triangle where the angles are in the ratio of 1:4:7, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Ratio**:
The angles of the triangle are given in the ratio 1:4:7. This means we can express the angles in terms of a variable \( x \).

Let:
- First angle = \( 1x \)
- Second angle = \( 4x \)
- Third angle = \( 7x \)

2. **Setting Up the Equation**:
We know that the sum of the angles in a triangle is always 180 degrees. Therefore, we can write the equation:
\[ 1x + 4x + 7x = 180 \]

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

4. **Solving for \( x \)**:
To find the value of \( x \), divide both sides of the equation by 12:
\[ x = \frac{180}{12} = 15 \]

5. **Finding Each Angle**:
Now that we have \( x = 15 \), we can find the measures of each angle:
- First angle = \( 1x = 1 \times 15 = 15 \) degrees
- Second angle = \( 4x = 4 \times 15 = 60 \) degrees
- Third angle = \( 7x = 7 \times 15 = 105 \) degrees

6. **Identifying the Largest Angle**:
Among the angles calculated, the largest angle is:
\[ 105 \text{ degrees} \]

### Conclusion:
The value of the largest angle in the triangle is **105 degrees**.

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