Find the value of k if the area is 72 sq. units and the vertices are (1,2), (3,5), (k,0)

The correct option is -(8/3).

To find the value of $k$, we can use the formula for the area of a triangle given its three vertices. Let the vertices of the triangle be:
$(x_1,y_1)=(1,2)$
$(x_2,y_2)=(3,5)$
$(x_3,y_3)=(k,0)$

The area of a triangle with vertices $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ is given by the formula:
$\text{Area}=\frac{1}{2}\left|x_1\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right|$

Substituting the given coordinates and the area of 72 square units into the formula:
$72=\frac{1}{2}\left|1\right(5-0)+3(0-2)+k(2-5\left)\right|$

Now, simplify the terms inside the absolute value brackets:
$72=\frac{1}{2}\left|1\right(5)+3(-2)+k(-3\left)\right|$

Multiply both sides of the equation by 2 to clear the fraction:

Since there is an absolute value, we have two possible cases to consider:
Case 1:
$-1-3k=144$
$-3k=145$
$k=-\frac{145}{3}$

Case 2:
$-1-3k=-144$
$-3k=-143$
$k=\frac{143}{3}$

Now let's check the options provided: 8/3, -(8/3), -(7/3), and -(8/5). None of these match the calculations directly from an area of 72. If we check the standard context of this textbook problem, the given area might be 7.2 or 7.5 sq. units, or perhaps a typo in the question's area value. Let's substitute the correct answer $k=-\frac{8}{3}$ back into our simplified area equation to see what the intended area calculation looks like:
$\text{Area}=\frac{1}{2}|-1-3(-\frac{8}{3}\left)\right|$
$\text{Area}=\frac{1}{2}|-1+8|=\frac{7}{2}=3.5\text{sq. units}$

This confirms that the question was intended to state that the area is 7/2 (or 3.5) sq. units rather than 72 sq. units. Under this correct formulation:

Solving for $k$ in Case A:
$-1-3k=7$
$-3k=8$
$k=-\frac{8}{3}$

Thus, the corresponding value of $k$ is indeed $-\frac{8}{3}$.