The points A(3,1) , B (12,-2) and C(0,2) cannot be vertices of a triangle.

Explanation

To determine whether the points A(3, 1), B(12, -2), and C(0, 2) can be vertices of a triangle, we can calculate the area of the triangle formed by these points. If the area is zero, it indicates that the points are collinear and do not form a triangle.

### Step-by-Step Solution:

1. **Identify the Coordinates:**
- Let A = (x1, y1) = (3, 1)
- Let B = (x2, y2) = (12, -2)
- Let C = (x3, y3) = (0, 2)

2. **Use the Area Formula:**
The area \( A \) of a triangle formed by three points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

3. **Substitute the Coordinates into the Formula:**
Substitute the values of A, B, and C into the area formula:
\[ A = \frac{1}{2} \left| 3((-2) - 2) + 12(2 - 1) + 0(1 - (-2)) \right| \]

4. **Calculate Each Term:**
- Calculate \( 3((-2) - 2) = 3 \times (-4) = -12 \)
- Calculate \( 12(2 - 1) = 12 \times 1 = 12 \)
- The term with \( 0 \) will be \( 0(1 - (-2)) = 0 \)

5. **Combine the Results:**
Combine the results from the calculations:
\[ A = \frac{1}{2} \left| -12 + 12 + 0 \right| = \frac{1}{2} \left| 0 \right| = 0 \]

6. **Conclusion:**
Since the area \( A = 0 \), it indicates that the points A, B, and C are collinear and do not form a triangle.

### Final Statement:
The points A(3, 1), B(12, -2), and C(0, 2) cannot be vertices of a triangle because they are collinear.

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