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To find the area of the quadrilateral formed by points \( E \), \( F \), \( G \), and \( H \), we can use the formula for the area of a polygon given its vertices' coordinates. The coordinates of each vertex are as follows:

- \( E (-3, 5) \)
- \( F (5, 2) \)
- \( G (4, -4) \)
- \( H (-5, -1) \)

The area of a polygon with vertices \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\) is calculated using the shoelace formula:

\[
\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + \dots + x_{n-1}y_n + x_ny_1 - (y_1x_2 + y_2x_3 + \dots + y_{n-1}x_n + y_nx_1) \right|
\]

Substitute in the coordinates of points \( E \), \( F \), \( G \), and \( H \):

\[
\text{Area} = \frac{1}{2} \left| (-3 \cdot 2) + (5 \cdot -4) + (4 \cdot -1) + (-5 \cdot 5) - \left(5 \cdot 5 + 2 \cdot 4 + (-4) \cdot -5 + (-1) \cdot -3\right) \right|
\]

Let's compute each term:

1. \( -3 \cdot 2 = -6 \)
2. \( 5 \cdot -4 = -20 \)
3. \( 4 \cdot -1 = -4 \)
4. \( -5 \cdot 5 = -25 \)

So, the first part of the formula (summing these products) gives:

\[
-6 + (-20) + (-4) + (-25) = -55
\]

Now for the second set of terms:

1. \( 5 \cdot 5 = 25 \)
2. \( 2 \cdot 4 = 8 \)
3. \( -4 \cdot -5 = 20 \)
4. \( -1 \cdot -3 = 3 \)

Summing these products gives:

\[
25 + 8 + 20 + 3 = 56
\]

Now, substitute these into the formula:

\[
\text{Area} = \frac{1}{2} \left| -55 - 56 \right| = \frac{1}{2} \times 111 = 55.5
\]

Rounding to the nearest whole number, the area of the quadrilateral is approximately:

\[
\text{Area} \approx 56 \text{ square units}
\]

Would you like further details or have any questions about this calculation?

Here are some additional questions to consider:
1. How would the formula change if there were more vertices in the polygon?
2. Can you verify the answer by dividing the quadrilateral into triangles and summing their areas?
3. How does the orientation of the polygon affect the calculation of its area?
4. What other methods can be used to calculate the area of an irregular quadrilateral?
5. How would the calculation change if the coordinates were all positive?

**Tip:** The shoelace formula is highly efficient for polygons; just make sure to list the vertices in order!

Find the area of the rectangle formed by points E, F, G, and H. Round to the nearest whole number.

Explore how to find the area of a quadrilateral formed by points on a coordinate plane using the shoelace formula. This example demonstrates each step, including coordinates for points E, F, G, and H, and rounding the result to the nearest whole number.

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