Three points A(1,-2) ,B(3,4) and C(4,7) form :

Options

A. a. a straight line
B. b. an equilateral triangle
C. c. a right angled triangle
D. d. none of (a) , (b) ,(c)

Answer

To determine the relationship among the points A(1, -2), B(3, 4), and C(4, 7), we will calculate the slopes of the lines formed by these points and analyze the results.

### Step-by-Step Solution:

1. **Identify the Points**:
- A(1, -2)
- B(3, 4)
- C(4, 7)

2. **Calculate the Slope of Line AB**:
- The formula for the slope (m) between two points (x1, y1) and (x2, y2) is:
\[ m = \frac{y2 - y1}{x2 - x1} \]
- For points A(1, -2) and B(3, 4):
\[ m_{AB} = \frac{4 - (-2)}{3 - 1} = \frac{4 + 2}{3 - 1} = \frac{6}{2} = 3 \]

3. **Calculate the Slope of Line BC**:
- For points B(3, 4) and C(4, 7):
\[ m_{BC} = \frac{7 - 4}{4 - 3} = \frac{3}{1} = 3 \]

4. **Compare the Slopes**:
- We found that:
\[ m_{AB} = 3 \quad \text{and} \quad m_{BC} = 3 \]
- Since \( m_{AB} = m_{BC} \), the lines AB and BC are parallel.

5. **Conclusion**:
- Since points A, B, and C are such that the lines connecting them (AB and BC) are parallel, it indicates that all three points lie on the same straight line. Therefore, the answer is that the points A, B, and C form a straight line.

### Final Answer:
The points A(1, -2), B(3, 4), and C(4, 7) form a **straight line**.