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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.

Learn how to calculate the area of triangle FDE using the coordinates of its vertices. This problem utilizes coordinate geometry and the formula for the area of a triangle. Suitable for students in Grades 8-10.

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