Check how the points A,B and C are situated where `A(4,0),B(-1,-1),C(3,5)` .

Explanation

To determine how the points A(4,0), B(-1,-1), and C(3,5) are situated, we will calculate the distances between each pair of points using the distance formula.

### Step-by-Step Solution

1. **Identify the Points**:
- A(4, 0)
- B(-1, -1)
- C(3, 5)

2. **Distance Formula**:
The distance \( D \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ D = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \]

3. **Calculate Distance AB**:
- Using points A(4, 0) and B(-1, -1):
\[ AB = \sqrt{(4 - (-1))^2 + (0 - (-1))^2} \]
\[ = \sqrt{(4 + 1)^2 + (0 + 1)^2} \]
\[ = \sqrt{5^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26} \]

4. **Calculate Distance BC**:
- Using points B(-1, -1) and C(3, 5):
\[ BC = \sqrt{(-1 - 3)^2 + (-1 - 5)^2} \]
\[ = \sqrt{(-4)^2 + (-6)^2} \]
\[ = \sqrt{16 + 36} = \sqrt{52} \]

5. **Calculate Distance AC**:
- Using points A(4, 0) and C(3, 5):
\[ AC = \sqrt{(4 - 3)^2 + (0 - 5)^2} \]
\[ = \sqrt{(1)^2 + (-5)^2} \]
\[ = \sqrt{1 + 25} = \sqrt{26} \]

6. **Summarize the Distances**:
- \( AB = \sqrt{26} \)
- \( BC = \sqrt{52} \)
- \( AC = \sqrt{26} \)

7. **Check the Triangle Inequality**:
- For points A, B, and C to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
- Check:
- \( AB + AC > BC \)
\[ \sqrt{26} + \sqrt{26} > \sqrt{52} \]
\[ 2\sqrt{26} > \sqrt{52} \]
- Since \( \sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13} \), we need to check if \( 2\sqrt{26} > 2\sqrt{13} \), which simplifies to \( \sqrt{26} > \sqrt{13} \). This is true since \( 26 > 13 \).

8. **Conclusion**:
- Since \( AB = AC \) and \( AB + AC > BC \), the triangle formed by points A, B, and C is an **Isosceles Triangle**.