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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.

The vertices of triangle DEF are D(-6, 2), E(3, 5), F(8, -7). Find the area of this triangle.

This problem involves finding the area of a triangle with given vertices D(-6, 2), E(3, 5), and F(8, -7) using the formula for the area of a triangle in the coordinate plane. The solution demonstrates step-by-step how to substitute the coordinates into the formula and simplify the expression.

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