To find the area of triangle DEF with vertices $D(4, -3)$, $E(-2, 1)$, and $F(-9, -6)$, you 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 in the coordinates:

• $x_1 = 4$, $y_1 = -3$
• $x_2 = -2$, $y_2 = 1$
• $x_3 = -9$, $y_3 = -6$

Calculating:

$\text{Area} = \frac{1}{2} \left| 4(1 - (-6)) + (-2)(-6 - (-3)) + (-9)(-3 - 1) \right|$

$= \frac{1}{2} \left| 4(7) + (-2)(-3) + (-9)(-4) \right|$

$= \frac{1}{2} \left| 28 + 6 + 36 \right| = \frac{1}{2} \left| 70 \right| = 35$

Thus, the area of triangle DEF is $35$ square units.

Would you like more details or have any questions? Here are some related questions to consider:

1. How would the area change if one vertex was moved?
2. What is the perimeter of triangle DEF?
3. How can the area be calculated using the base and height?
4. What type of triangle is DEF (scalene, isosceles, or equilateral)?
5. Can you find the centroid of triangle DEF?

Tip: Always double-check your calculations to avoid minor errors!