To solve the problem of finding the area of a triangle given three points on a plane, we can use the shoelace formula (derived from the cross product of vectors), which is efficient and straightforward for coordinate-based calculations.

Key Formula

For three points (A(x_1,y_1)), (B(x_2,y_2)), (C(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| ]

The absolute value ensures the area is positive, and multiplying by (\frac{1}{2}) accounts for the triangle being half the area of the parallelogram formed by vectors (\overrightarrow{AB}) and (\overrightarrow{AC}).

Step-by-Step Example

Suppose the three points are (A(1,2)), (B(3,4)), (C(5,0)):

1. Substitute into the formula:
   [ \text{Area} = \frac{1}{2} \left| 1(4-0) + 3(0-2) +5(2-4) \right| ]

2. Compute the expression inside the absolute value:
   [ 1(4) +3(-2) +5(-2) =4 -6 -10 = -12 ]

3. Take absolute value and multiply by (\frac{1}{2}):
   [ \text{Area} = \frac{1}{2} \times |-12| =6 ]

Note

If the result is zero, the three points are collinear (no triangle exists).

Apply this formula to your specific three points to get the area! If you provide the coordinates of the points, I can compute the exact area for you.

Answer: (Depends on the specific points; for the example above, it's 6. Replace with your calculation using the formula.)
(\boxed{6}) (example value—adjust based on your points)

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