The problem asks for the area of triangle $\triangle DEF$ with vertices at the points:

$D(-6, 2), E(3, 5), F(8, -7)$

To find the area of a triangle with given vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, we can use the formula:

$\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$

Substituting the points $D(-6, 2)$, $E(3, 5)$, and $F(8, -7)$:

$\text{Area} = \frac{1}{2} \left| -6(5 - (-7)) + 3((-7) - 2) + 8(2 - 5) \right|$

Now, simplifying:

$= \frac{1}{2} \left| -6(12) + 3(-9) + 8(-3) \right|$ $= \frac{1}{2} \left| -72 - 27 - 24 \right|$ $= \frac{1}{2} \left| -123 \right|$ $= \frac{1}{2} \times 123 = 61.5$

Thus, the area of triangle $\triangle DEF$ is 61.5 square units, which corresponds to option D.

Would you like more details on any step? Here are 5 questions that expand on this topic:

1. What is the geometric significance of the area formula for triangles with coordinates?
2. How can you verify the correctness of the area calculated for a triangle?
3. What other methods can be used to find the area of a triangle with vertices in the plane?
4. Can this formula be extended to polygons with more than three vertices?
5. How does this area formula relate to vector cross products?

Tip: Always double-check signs when applying this area formula, as one small mistake in sign can lead to an incorrect result.