To find the area of the triangle with vertices at points \( C(6,2) \), \( B(4,6) \), and \( A(1,2) \), we can use the formula for the area of a triangle given by the coordinates of its vertices:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Let the coordinates of the vertices be:
- \( A(1, 2) \) has \( x_1 = 1 \), \( y_1 = 2 \)
- \( B(4, 6) \) has \( x_2 = 4 \), \( y_2 = 6 \)
- \( C(6, 2) \) has \( x_3 = 6 \), \( y_3 = 2 \)
Now, substituting these coordinates into the formula:
\[ \text{Area} = \frac{1}{2} \left| 1(6 - 2) + 4(2 - 2) + 6(2 - 6) \right| \]
Calculating each term:
- \( 1(6 - 2) = 1 \cdot 4 = 4 \)
- \( 4(2 - 2) = 4 \cdot 0 = 0 \)
- \( 6(2 - 6) = 6 \cdot -4 = -24 \)
Now putting it all together:
\[ \text{Area} = \frac{1}{2} \left| 4 + 0 - 24 \right| = \frac{1}{2} \left| 4 - 24 \right| = \frac{1}{2} \left| -20 \right| = \frac{1}{2} \cdot 20 = 10 \]
Thus, the area of Fiona's triangle is
\[ \boxed{10} \quad \text{square units.} \]
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