If the angles of a triangle are in the ratio of 2 : 3 : 7 , then find the ratio of the greatest angle to the smallest angle
A. 7 : 2
B. 2 : 3
C. 7 : 1
D. 3 : 5

Options

A. C
B. A
C. B
D. D

Answer

To solve the problem, we need to find the angles of a triangle that are in the ratio of 2:3:7 and then determine the ratio of the greatest angle to the smallest angle.

### Step-by-Step Solution:

1. **Understanding the Ratio**:
The angles of the triangle are given in the ratio of 2:3:7. We can express these angles in terms of a variable \( x \):
- Let the angles be \( 2x \), \( 3x \), and \( 7x \).

2. **Sum of Angles in a Triangle**:
The sum of the angles in any triangle is always \( 180^\circ \). Therefore, we can set up the equation:
\[ 2x + 3x + 7x = 180^\circ \]

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

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

5. **Calculate the Angles**:
Now substitute \( x \) back to find the actual angles:
- The first angle: \( 2x = 2 \times 15^\circ = 30^\circ \)
- The second angle: \( 3x = 3 \times 15^\circ = 45^\circ \)
- The third angle: \( 7x = 7 \times 15^\circ = 105^\circ \)

6. **Identify the Greatest and Smallest Angles**:
From the calculated angles:
- Smallest angle: \( 30^\circ \)
- Greatest angle: \( 105^\circ \)

7. **Find the Ratio of the Greatest Angle to the Smallest Angle**:
The ratio of the greatest angle to the smallest angle is:
\[ \text{Ratio} = \frac{105^\circ}{30^\circ} = \frac{105}{30} = \frac{7}{2} \]

### Conclusion:
The ratio of the greatest angle to the smallest angle is \( 7:2 \).

### Final Answer:
**A. 7 : 2**