Find the area of the triangle with vertices at the points given in each of the following . Are the following points collinear ?
`(-2,-3)`, `(3,2)`, `(-1,-8)`

Answer

To find the area of the triangle formed by the vertices at the points \((-2,-3)\), \((3,2)\), and \((-1,-8)\), we can use the formula for the area of a triangle given by the coordinates of its vertices:

\[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \]

Where:
- \((x_1, y_1) = (-2, -3)\)
- \((x_2, y_2) = (3, 2)\)
- \((x_3, y_3) = (-1, -8)\)

### Step 1: Substitute the coordinates into the formula

Substituting the values into the area formula:

\[ \text{Area} = \frac{1}{2} \left| -2(2 - (-8)) + 3((-8) - (-3)) + (-1)((-3) - 2) \right| \]

### Step 2: Simplify the expressions inside the absolute value

Calculating each term inside the absolute value:

1. For the first term:
\[ -2(2 + 8) = -2 \times 10 = -20 \]

2. For the second term:
\[ 3(-8 + 3) = 3 \times (-5) = -15 \]

3. For the third term:
\[ -1(-3 - 2) = -1 \times (-5) = 5 \]

### Step 3: Combine the terms

Now, combine all the terms:

\[ \text{Area} = \frac{1}{2} \left| -20 - 15 + 5 \right| = \frac{1}{2} \left| -30 \right| \]

### Step 4: Calculate the absolute value and the area

The absolute value of \(-30\) is \(30\):

\[ \text{Area} = \frac{1}{2} \times 30 = 15 \]

### Conclusion

Thus, the area of the triangle is:

\[ \text{Area} = 15 \text{ square units} \]

### Step 5: Check for collinearity

To check if the points are collinear, we look at the area calculated. If the area is zero, the points are collinear. Since the area is \(15\), which is not zero, the points are not collinear.

### Final Answer

- Area of the triangle: \(15\) square units
- The points are not collinear.