Name the type of quadrilateral formed, if any, by the following points and give reasona for your Solution :
(-3,5), (3,1), (0,3), (-1,-4)

Answer

To determine the type of quadrilateral formed by the points (-3, 5), (3, 1), (0, 3), and (-1, -4), we will follow these steps:

### Step 1: Identify the points
Let's label the points as follows:
- A = (-3, 5)
- B = (3, 1)
- C = (0, 3)
- D = (-1, -4)

### Step 2: Use the distance formula
The distance formula between two points (x1, y1) and (x2, y2) is given by:
\[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]

### Step 3: Calculate the lengths of the sides
1. **Calculate AB**:
\[ AB = \sqrt{(3 - (-3))^2 + (1 - 5)^2} = \sqrt{(3 + 3)^2 + (1 - 5)^2} = \sqrt{(6)^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} \]

2. **Calculate BC**:
\[ BC = \sqrt{(0 - 3)^2 + (3 - 1)^2} = \sqrt{(-3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13} \]

3. **Calculate CD**:
\[ CD = \sqrt{(-1 - 0)^2 + (-4 - 3)^2} = \sqrt{(-1)^2 + (-7)^2} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2} \]

4. **Calculate DA**:
\[ DA = \sqrt{(-3 - (-1))^2 + (5 - (-4))^2} = \sqrt{(-2)^2 + (9)^2} = \sqrt{4 + 81} = \sqrt{85} \]

### Step 4: Compare the lengths
Now we have the lengths of all four sides:
- AB = \(2\sqrt{13}\)
- BC = \(\sqrt{13}\)
- CD = \(5\sqrt{2}\)
- DA = \(\sqrt{85}\)

Since all four sides are of different lengths, we conclude that the quadrilateral is not a rhombus, rectangle, or square.

### Step 5: Conclusion
Since none of the sides are equal and there are no special properties (like parallel sides), the quadrilateral formed by the points (-3, 5), (3, 1), (0, 3), and (-1, -4) is a **general quadrilateral**.

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