What Are the Possible Values of c for a Triangle with Area 5?

[anemone]

Here is this week's POTW:

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A triangle with vertices $A(0,\,0),\,B(3,\,4)$ and $C(2,\,c)$ has area 5 units². Find all possible values of $c$.

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[anemone]

Congratulations to skeeter for his correct solution, which you can find below:

Spoiler

for $0 \le c \le 4$, maximum area = 4

for $c > 4$ ...

$\vec{AB} \times \vec{AC} = 10$

\[ \begin{vmatrix} 1 & 1 & 1\\ 3 & 4 & 0\\ 2 & c & 0 \end{vmatrix}=10 \implies 3c-8 = 10 \implies c=6 \]

for $c < 0$ ...

$\vec{AC} \times \vec{AB} = 10$

\[ \begin{vmatrix} 1 & 1 & 1\\ 2 & c & 0\\ 3 & 4 & 0 \end{vmatrix}=10 \implies 8-3c = 10 \implies c=-\dfrac{2}{3} \]