Determine whether the points are collinear OR not
`A(1, -2), B(2, -5), C(-4, 7)`

Answer

To determine whether the points A(1, -2), B(2, -5), and C(-4, 7) are collinear, we can use the concept of distances between the points. If the sum of the distances between any two points equals the distance between the third point, then the points are collinear.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let A(1, -2), B(2, -5), and C(-4, 7).

2. **Calculate the Distance AB**:
- The formula for the distance between two points (x1, y1) and (x2, y2) is:
\[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]
- For points A(1, -2) and B(2, -5):
\[ d_{AB} = \sqrt{(2 - 1)^2 + (-5 - (-2))^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]

3. **Calculate the Distance BC**:
- For points B(2, -5) and C(-4, 7):
\[ d_{BC} = \sqrt{(-4 - 2)^2 + (7 - (-5))^2} = \sqrt{(-6)^2 + (12)^2} = \sqrt{36 + 144} = \sqrt{180} = 6\sqrt{5} \]

4. **Calculate the Distance AC**:
- For points A(1, -2) and C(-4, 7):
\[ d_{AC} = \sqrt{(-4 - 1)^2 + (7 - (-2))^2} = \sqrt{(-5)^2 + (9)^2} = \sqrt{25 + 81} = \sqrt{106} \]

5. **Check for Collinearity**:
- For the points to be collinear, the sum of the distances between any two points should equal the distance between the third point.
- Check if \( d_{AB} + d_{BC} = d_{AC} \):
\[ \sqrt{10} + 6\sqrt{5} \neq \sqrt{106} \]
- Since this condition does not hold true, the points A, B, and C are **not collinear**.

### Conclusion:
The points A(1, -2), B(2, -5), and C(-4, 7) are **not collinear**.