The area of a triangle with vertices $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$ is given by the determinant formula: $$Area = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$$
Given vertices are $A(3, 1)$, $B(-2, 1)$, and $C(0, k)$. The area is 5 sq. units. $$5 = \frac{1}{2} |3(1 - k) + (-2)(k - 1) + 0(1 - 1)|$$
Multiply by 2 and simplify the terms inside the modulus: $$10 = |3 - 3k - 2k + 2|$$ $$10 = |5 - 5k|$$
This implies two cases: Case 1: $5 - 5k = 10 \implies -5k = 5 \implies k = -1$ Case 2: $5 - 5k = -10 \implies -5k = -15 \implies k = 3$
Final Answer: -1, 3
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