To find the area of the triangle $\triangle FDE$, we need to determine the coordinates of the vertices $F$, $D$, and $E$, and apply the formula for the area of a triangle using its vertices in the coordinate plane.

The area $A$ of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by:

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

Step 1: Identify the coordinates of $F$, $D$, and $E$ from the graph.

• Point $F$ is at $(4, 8)$.
• Point $D$ is at $(10, 4)$.
• Point $E$ is at $(8, 10)$.

Step 2: Apply the formula.

Using the coordinates $F(4, 8)$, $D(10, 4)$, and $E(8, 10)$:

$A = \frac{1}{2} \left| 4(4 - 10) + 10(10 - 8) + 8(8 - 4) \right|$

Now, I will compute the value.The area of the triangle $\triangle FDE$ is 14 square units.

Would you like more details or have any questions about this process? Here are five related questions to explore further:

1. How can we calculate the perimeter of triangle $\triangle FDE$?
2. How do we determine whether a triangle is right-angled based on its vertices?
3. What is the significance of using absolute value in the triangle area formula?
4. How does the formula for the area of a triangle extend to higher dimensions?
5. Can we calculate the area of a polygon with more than three sides using a similar formula?

Tip: When calculating the area of a triangle using coordinates, make sure you arrange the points in order, either clockwise or counterclockwise, to apply the formula correctly.