To solve the problem, let's analyze the given information and find the possible locations of point $F$ such that the triangle $\Delta DEF$ has an area of 28 square units.

Step 1: Calculate the Length of Segment $DE$

The coordinates of points $D$ and $E$ are $D(-6, 3)$ and $E(2, 3)$, respectively. Since both points have the same y-coordinate, the segment $DE$ is horizontal.

The length of $DE$ can be found using the distance formula for horizontal segments: $\text{Length of } DE = |x_2 - x_1| = |2 - (-6)| = 2 + 6 = 8 \text{ units}$

Step 2: Use the Area Formula for a Triangle

The area of a triangle can be calculated using the formula: $\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}$ Given that the area is 28 square units and the base $DE$ is 8 units: $28 = \frac{1}{2} \times 8 \times \text{Height}$ $28 = 4 \times \text{Height}$ $\text{Height} = \frac{28}{4} = 7 \text{ units}$ The height is the perpendicular distance from point $F$ to the line $DE$.

Step 3: Determine the Coordinates of $F$

The height must be 7 units above or below the line $DE$, which has a y-coordinate of 3. Therefore, the possible y-coordinates of $F$ are: $y = 3 + 7 = 10 \quad \text{or} \quad y = 3 - 7 = -4$

Step 4: Check the Possible Points

We now check the provided options:

• (7, 4): The y-coordinate is 4, which doesn't match the possible heights (10 or -4). Not a valid point.
• (1, 10): The y-coordinate is 10, which is 7 units above the line $DE$. Valid point.
• (-6, 10): The y-coordinate is 10, which is 7 units above the line $DE$. Valid point.
• (5, -4): The y-coordinate is -4, which is 7 units below the line $DE$. Valid point.

Conclusion:

The valid points for $F$ where the area of $\Delta DEF$ equals 28 square units are:

• (1, 10)
• (-6, 10)
• (5, -4)

The selected answers are correct.

Would you like more details or have any questions?

Here are five related questions for further practice:

1. How would the area change if $F$ were at a different height?
2. What would happen if the base $DE$ were longer or shorter?
3. How would you find the area of $\Delta DEF$ if $F$ had different coordinates?
4. What if the triangle had a different area, say 56 square units?
5. Can you calculate the perimeter of $\Delta DEF$ with the selected points $F$?

Tip: Remember that the height in the area formula for a triangle always refers to the perpendicular distance from a vertex to the opposite side.