Determine the area of the quadrilateral whose vertices are (-14, -2), (10, 10), (-10,-10), and (4, -24).

To determine the area of a quadrilateral given its vertices, we can use the formula derived from the Shoelace Theorem (or Gauss's area formula). The formula for the area \( A \) of a quadrilateral with vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\), and \((x_4, y_4)\) is:

\[
A = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right|
\]

Given the vertices:
\[
(x_1, y_1) = (-14, -2)
\]
\[
(x_2, y_2) = (10, 10)
\]
\[
(x_3, y_3) = (-10, -10)
\]
\[
(x_4, y_4) = (4, -24)
\]

We substitute these values into the formula:

\[
A = \frac{1}{2} \left| (-14)(10) + (10)(-10) + (-10)(-24) + (4)(-2) - \left[ (-2)(10) + (10)(-10) + (-10)(4) + (-24)(-14) \right] \right|
\]

Let's calculate each term step by step:

1. **First sum of products:**
\[
-14 \times 10 = -140
\]
\[
10 \times -10 = -100
\]
\[
-10 \times -24 = 240
\]
\[
4 \times -2 = -8
\]

So, the first part of the formula is:
\[
-140 - 100 + 240 - 8 = -8
\]

2. **Second sum of products:**
\[
-2 \times 10 = -20
\]
\[
10 \times -10 = -100
\]
\[
-10 \times 4 = -40
\]
\[
-24 \times -14 = 336
\]

So, the second part of the formula is:
\[
-20 - 100 - 40 + 336 = 176
\]

Now, subtract the second sum from the first:

\[
-8 - 176 = -184
\]

Finally, apply the absolute value and the \(\frac{1}{2}\) factor:

\[
A = \frac{1}{2} \times | -184 | = \frac{1}{2} \times 184 = 92
\]

So, the area of the quadrilateral is \( \boxed{92} \) square units.

Would you like more details on the steps or any other questions? 

Here are 5 questions related to this problem:

1. How does the Shoelace Theorem relate to the determinant of a matrix?
2. What is the significance of the absolute value in the area formula?
3. Can this method be applied to polygons with more than four sides?
4. How would the method change if the vertices were given in a different order?
5. How does the Shoelace Theorem work for concave quadrilaterals?

**Tip:** Always ensure to follow the vertex order (either clockwise or counterclockwise) consistently when applying the Shoelace Theorem to avoid sign errors.

Learn how to calculate the area of a quadrilateral with vertices at (-14, -2), (10, 10), (-10, -10), and (4, -24) using the Shoelace Theorem. Detailed step-by-step solution provided.

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