Find the area of the triangle with vertices at the points given in each of the following . Are the following points collinear ?
`(0,0),(6,0),(4,3)`

Answer

To find the area of the triangle formed by the vertices at the points (0,0), (6,0), and (4,3), we can use the formula for the area of a triangle given by its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

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

### Step 1: Assign the coordinates to the variables
Let:
- \( (x_1, y_1) = (0, 0) \)
- \( (x_2, y_2) = (6, 0) \)
- \( (x_3, y_3) = (4, 3) \)

### Step 2: Substitute the coordinates into the area formula
Substituting the values into the area formula:

\[ \text{Area} = \frac{1}{2} \left| 0(0 - 3) + 6(3 - 0) + 4(0 - 0) \right| \]

### Step 3: Simplify the expression
Calculating each term:

- The first term: \( 0(0 - 3) = 0 \)
- The second term: \( 6(3 - 0) = 18 \)
- The third term: \( 4(0 - 0) = 0 \)

Thus, we have:

\[ \text{Area} = \frac{1}{2} \left| 0 + 18 + 0 \right| = \frac{1}{2} \left| 18 \right| = \frac{18}{2} = 9 \]

### Step 4: Conclusion about the area and collinearity
The area of the triangle is \( 9 \) square units. Since the area is not zero, the points are not collinear.

### Final Answer
- The area of the triangle is \( 9 \) square units.
- The points are **not collinear**.

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