The ratio of angles of a quadrilateral is 1 : 2 · 3 · 4 . What is the sum of twice the second smallest angle and half of the largest angle?

Options

A. `72^@`
B. `144^@`
C. `108^(@)`
D. `216^@`

Answer

To solve the question, we need to find the angles of the quadrilateral based on the given ratio and then calculate the required sum of twice the second smallest angle and half of the largest angle.

### Step-by-Step Solution:

1. **Understand the Ratio**: The angles of the quadrilateral are in the ratio 1 : 2 : 3 : 4. We can represent the angles as:
- First angle = 1x
- Second angle = 2x
- Third angle = 3x
- Fourth angle = 4x

2. **Sum of Angles in a Quadrilateral**: The sum of the angles in any quadrilateral is 360 degrees. Therefore, we can set up the equation:
\[ 1x + 2x + 3x + 4x = 360 \]

3. **Combine Like Terms**: Simplifying the left side, we have:
\[ 10x = 360 \]

4. **Solve for x**: To find the value of x, divide both sides by 10:
\[ x = \frac{360}{10} = 36 \]

5. **Calculate Each Angle**:
- First angle = \(1x = 1 \times 36 = 36\) degrees
- Second angle = \(2x = 2 \times 36 = 72\) degrees
- Third angle = \(3x = 3 \times 36 = 108\) degrees
- Fourth angle = \(4x = 4 \times 36 = 144\) degrees

6. **Identify the Angles**: The angles of the quadrilateral are:
- 36 degrees (smallest)
- 72 degrees (second smallest)
- 108 degrees
- 144 degrees (largest)

7. **Calculate Twice the Second Smallest Angle**: The second smallest angle is 72 degrees. Therefore, twice this angle is:
\[ 2 \times 72 = 144 \text{ degrees} \]

8. **Calculate Half of the Largest Angle**: The largest angle is 144 degrees. Therefore, half of this angle is:
\[ \frac{144}{2} = 72 \text{ degrees} \]

9. **Sum the Results**: Now, we add twice the second smallest angle and half of the largest angle:
\[ 144 + 72 = 216 \text{ degrees} \]

### Final Answer:
The sum of twice the second smallest angle and half of the largest angle is **216 degrees**.