To calculate the area of a triangle given its vertices, you can use the following formula based on the coordinates of the vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \]
Given the vertices of the triangle:
- \( A (-7, 3) \)
- \( B (5, 8) \)
- \( C (0, -2) \)
Let's assign the coordinates:
- \( (x_1, y_1) = (-7, 3) \)
- \( (x_2, y_2) = (5, 8) \)
- \( (x_3, y_3) = (0, -2) \)
Plugging in the values:
\[ \text{Area} = \frac{1}{2} \left| -7(8 - (-2)) + 5((-2) - 3) + 0(3 - 8) \right| \]
Calculating each term:
- \( -7(8 + 2) = -7 \times 10 = -70 \)
- \( 5(-2 - 3) = 5 \times -5 = -25 \)
- \( 0(3 - 8) = 0 \)
Now substitute these into the formula:
\[ \text{Area} = \frac{1}{2} \left| -70 - 25 + 0 \right| \]
This simplifies to:
\[ \text{Area} = \frac{1}{2} \left| -95 \right| = \frac{1}{2} \times 95 = 47.5 \]
Thus, the area of the triangle is \( \boxed{47.5} \) square units.
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