To calculate the area of the triangle with vertices at points \( A(6, 2) \), \( B(4, 6) \), and \( C(1, 2) \), we can use the formula for the area of a triangle given by its vertices in the Cartesian coordinate system:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Here, we have:
- \( (x_1, y_1) = (6, 2) \)
- \( (x_2, y_2) = (4, 6) \)
- \( (x_3, y_3) = (1, 2) \)
Substituting the coordinates into the formula:
\[ \text{Area} = \frac{1}{2} \left| 6(6 - 2) + 4(2 - 2) + 1(2 - 6) \right| \]
Calculating each term step-by-step:
- \( 6(6 - 2) = 6 \times 4 = 24 \)
- \( 4(2 - 2) = 4 \times 0 = 0 \)
- \( 1(2 - 6) = 1 \times -4 = -4 \)
Now, substituting these values back into the area formula:
\[ \text{Area} = \frac{1}{2} \left| 24 + 0 - 4 \right| \]
Simplifying:
\[ \text{Area} = \frac{1}{2} \left| 20 \right| = \frac{1}{2} \times 20 = 10 \]
Thus, the area of Fiona's triangle is
\[ \boxed{10} \text{ square units.} \]
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