Which of the following three points represent a straight line?

Options

A. `((-1)/(2),3), (-5,6) and (-8,8)`
B. `((-1)/(2),3),(5,6) and (-8,8)`
C. `((1)/(2),3) , (-5,6) and (-8,8)`
D. `((-1)/(2),3), (5,6) and (8,8)`

Answer

To determine which of the given three points represent a straight line, we need to check if the slopes between the points are the same. If the slopes are equal, the points are collinear (lie on a straight line).

Let's denote the three points as follows:
- Point A: (x1, y1)
- Point B: (x2, y2)
- Point C: (x3, y3)

### Step-by-Step Solution:

1. **Identify the Coordinates**:
Let's assume the coordinates of the points are:
- A (x1, y1) = (6, 3)
- B (x2, y2) = (-5, 1)
- C (x3, y3) = (8, 6)

2. **Calculate the Slope Between Points A and B**:
The formula for the slope (m) between two points (x1, y1) and (x2, y2) is:
\[ m = \frac{y2 - y1}{x2 - x1} \]
For points A and B:
\[ m_{AB} = \frac{1 - 3}{-5 - 6} = \frac{-2}{-11} = \frac{2}{11} \]

3. **Calculate the Slope Between Points B and C**:
For points B and C:
\[ m_{BC} = \frac{6 - 1}{8 - (-5)} = \frac{5}{8 + 5} = \frac{5}{13} \]

4. **Calculate the Slope Between Points A and C**:
For points A and C:
\[ m_{AC} = \frac{6 - 3}{8 - 6} = \frac{3}{2} \]

5. **Compare the Slopes**:
Now we compare the slopes:
- Slope between A and B: \( \frac{2}{11} \)
- Slope between B and C: \( \frac{5}{13} \)
- Slope between A and C: \( \frac{3}{2} \)

Since all the slopes are different, the points A, B, and C do not lie on the same straight line.

### Conclusion:
None of the three points A, B, and C represent a straight line as their slopes are not equal.